INTERNATIONAL GONGRESS . For each such mmmmm, a straightforward application of the Chinese Remainder Theorem gives that the number of admissible pairs (a,b)(a,b)(a,b)(a, b)(a,b) is
The next step is to study averages of exp(∑p∣qr/gdd(q,r)2,p>M(q,r)1/p)expâ¡âˆ‘p∣qr/gddâ¡(q,r)2,p>M(q,r) 1/pexp(sum_(p∣qr//gdd(q,r)^(2),p > M(q,r))1//p)\exp \left(\sum_{p \mid q r / \operatorname{gdd}(q, r)^{2}, p>M(q, r)} 1 / p\right)expâ¡(∑p∣qr/gddâ¡(q,r)2,p>M(q,r)1/p). This gets a bit too technical in general, so we focus on the following special case:
In particular, if Aq∗Aq∗A_(q)^(**)\mathcal{A}_{q}^{*}Aq∗ is as in (2.5) with Δq=1/(qN)Δq=1/(qN)Delta_(q)=1//(qN)\Delta_{q}=1 /(q N)Δq=1/(qN), then meas (⋃q∈SAq∗)≫1⋃q∈S Aq∗≫1(uuu_(q in S)A_(q)^(**))≫1\left(\bigcup_{q \in S} \mathcal{A}_{q}^{*}\right) \gg 1(⋃q∈SAq∗)≫1.
When N≫QN≫QN≫QN \gg QN≫Q, Theorem 3.5 follows from the work of ErdÅ‘Ìs and Vaaler (Theorem 2.12), but when N=o(Q)N=o(Q)N=o(Q)N=o(Q)N=o(Q) it was not known prior to [20] in this generality. The proof begins by adapting the ErdÅ‘s-Vaaler argument to this more general setup.
We study the contribution of such pairs to the left-hand side of (3.5): if (q,r)∈Btj+1(q,r)∈Btj+1(q,r)inB_(t_(j+1))(q, r) \in \mathscr{B}_{t_{j+1}}(q,r)∈Btj+1, then
Now, let us consider the special case when N≫QN≫QN≫QN \gg QN≫Q, which corresponds to the ErdÅ‘s-Vaaler theorem. The inequality gcd(q,r)>Q/(Nt)gcdâ¡(q,r)>Q/(Nt)gcd(q,r) > Q//(Nt)\operatorname{gcd}(q, r)>Q /(N t)gcdâ¡(q,r)>Q/(Nt) is then basically trivially, so we
must prove (3.6) by exploiting the condition Lt(q,r)>100Lt(q,r)>100L_(t)(q,r) > 100L_{t}(q, r)>100Lt(q,r)>100. Indeed, writing d=gcd(q,r)d=gcdâ¡(q,r)d=gcd(q,r)d=\operatorname{gcd}(q, r)d=gcdâ¡(q,r), q=dq1q=dq1q=dq_(1)q=d q_{1}q=dq1 and r=dr1r=dr1r=dr_(1)r=d r_{1}r=dr1, we find that λt(q1)>50λtq1>50lambda_(t)(q_(1)) > 50\lambda_{t}\left(q_{1}\right)>50λt(q1)>50 or λt(r1)>50λtr1>50lambda_(t)(r_(1)) > 50\lambda_{t}\left(r_{1}\right)>50λt(r1)>50. By symmetry, we have
Since the condition that Lt(q,r)>100Lt(q,r)>100L_(t)(q,r) > 100L_{t}(q, r)>100Lt(q,r)>100 is insufficient, let us ignore it temporarily and focus on the condition that gcd(q,r)>Q/(Nt)gcdâ¡(q,r)>Q/(Nt)gcd(q,r) > Q//(Nt)\operatorname{gcd}(q, r)>Q /(N t)gcdâ¡(q,r)>Q/(Nt) for all (q,r)∈Bt(q,r)∈Bt(q,r)inB_(t)(q, r) \in \mathscr{B}_{t}(q,r)∈Bt. There is an obvious way in which this condition can be satisfied for many pairs (q,r)∈S×S(q,r)∈S×S(q,r)in S xx S(q, r) \in S \times S(q,r)∈S×S, namely if there is some fixed integer d>Q/(Nt)d>Q/(Nt)d > Q//(Nt)d>Q /(N t)d>Q/(Nt) that divides a large proportion of integers in SSSSS. Notice that the number of total multiples of ddddd in [Q,2Q][Q,2Q][Q,2Q][Q, 2 Q][Q,2Q] is about Q/d<t⋅NQ/d<tâ‹…NQ//d < t*NQ / d<t \cdot NQ/d<tâ‹…N, which is compatible with (3.4). We thus arrive at the following key question:
It turns out that the answer to the Model Problem as stated is negative. However, a technical variant of it is true, that takes into account the weights φ(q)/qφ(q)/qvarphi(q)//q\varphi(q) / qφ(q)/q in (3.4) and (3.6), and that is asymmetric in qqqqq and rrrrr. We shall explain this in the next section.
To locate this "structured" subgraph G′G′G^(')G^{\prime}G′, we use an iterative "compression" argument, roughly inspired by the papers of Erdós-Ko-Rado [11] and Dyson [9]. With each iteration, we pass to a smaller set of vertices, where we have additional information about which primes divide them. Of course, we must ensure that we end up with a sizeable graph. We do this by judiciously choosing the new graph at each step so that it has at least as many edges as what the qualitative parameters of the old graph might naively suggest. This way the new graph will have improved "structure" and "quality." When the algorithm terminates, we will end up with a fully structured subset of SSSSS, where we know that all large GCDs are due to a large fixed common factor. This will then allow us to exploit the condition that Lt(q,r)>100Lt(q,r)>100L_(t)(q,r) > 100L_{t}(q, r)>100Lt(q,r)>100 for all edges (q,r)(q,r)(q,r)(q, r)(q,r). Importantly, our algorithm will also control the set BtBtB_(t)\mathscr{B}_{t}Bt in terms of the terminal edge set. Hence the savings from the condition Lt(q,r)>100Lt(q,r)>100L_(t)(q,r) > 100L_{t}(q, r)>100Lt(q,r)>100 in the terminal graph will be transferred to BtBtB_(t)\mathscr{B}_{t}Bt, establishing (3.6).
One technical complication is that the iterative algorithm necessitates to view GGGGG as a bipartite graph. In addition, it is important to use the weights φ(q)/qφ(q)/qvarphi(q)//q\varphi(q) / qφ(q)/q. We thus set
μ(V)=∑v∈Vφ(v)v for V⊂N;μ(E)=∑(v,w)∈Eφ(v)φ(w)vw for E⊂N2μ(V)=∑v∈V φ(v)v for V⊂N;μ(E)=∑(v,w)∈E φ(v)φ(w)vw for E⊂N2mu(V)=sum_(v inV)(varphi(v))/(v)quad" for "VsubN;quad mu(E)=sum_((v,w)inE)(varphi(v)varphi(w))/(vw)quad" for "EsubN^(2)\mu(\mathcal{V})=\sum_{v \in \mathcal{V}} \frac{\varphi(v)}{v} \quad \text { for } \mathcal{V} \subset \mathbb{N} ; \quad \mu(\mathcal{E})=\sum_{(v, w) \in \mathcal{E}} \frac{\varphi(v) \varphi(w)}{v w} \quad \text { for } \mathcal{E} \subset \mathbb{N}^{2}μ(V)=∑v∈Vφ(v)v for V⊂N;μ(E)=∑(v,w)∈Eφ(v)φ(w)vw for E⊂N2
Let us now explain the algorithm in more detail. We set V0=W0=ςV0=W0=Ï‚V_(0)=W_(0)=Ï‚\mathcal{V}_{0}=\mathcal{W}_{0}=\varsigmaV0=W0=Ï‚ and construct two decreasing sequences of sets V0⊇V1⊇V2⊇⋯V0⊇V1⊇V2⊇⋯V_(0)supeV_(1)supeV_(2)supe cdots\mathcal{V}_{0} \supseteq \mathcal{V}_{1} \supseteq \mathcal{V}_{2} \supseteq \cdotsV0⊇V1⊇V2⊇⋯ and W0⊇W1⊇W2⊇⋯W0⊇W1⊇W2⊇⋯W_(0)supeW_(1)supeW_(2)supe cdots\mathcal{W}_{0} \supseteq \mathcal{W}_{1} \supseteq \mathcal{W}_{2} \supseteq \cdotsW0⊇W1⊇W2⊇⋯, as well as a sequence of distinct primes p1,p2,…p1,p2,…p_(1),p_(2),dotsp_{1}, p_{2}, \ldotsp1,p2,… such that either pjpjp_(j)p_{j}pj divides all elements of VjVjV_(j)\mathcal{V}_{j}Vj, or it is coprime to all elements of VjVjV_(j)\mathcal{V}_{j}Vj (and similarly with WjWjW_(j)\mathcal{W}_{j}Wj ). Since SSSSS consists solely of square-free integers, there are integers aj,bjaj,bja_(j),b_(j)a_{j}, b_{j}aj,bj dividing p1⋯pjp1⋯pjp_(1)cdotsp_(j)p_{1} \cdots p_{j}p1⋯pj, and such that gcd(v,p1⋯pj)=ajgcdâ¡v,p1⋯pj=ajgcd(v,p_(1)cdotsp_(j))=a_(j)\operatorname{gcd}\left(v, p_{1} \cdots p_{j}\right)=a_{j}gcdâ¡(v,p1⋯pj)=aj and gcd(w,p1⋯pj)=bjgcdâ¡w,p1⋯pj=bjgcd(w,p_(1)cdotsp_(j))=b_(j)\operatorname{gcd}\left(w, p_{1} \cdots p_{j}\right)=b_{j}gcdâ¡(w,p1⋯pj)=bj for all v∈Vjv∈Vjv inV_(j)v \in \mathcal{V}_{j}v∈Vj and all w∈Wjw∈Wjw inW_(j)w \in \mathcal{W}_{j}w∈Wj.
Let us now explain how to make the choice of which subgraph to focus on each time. Let Gj=(Vj,Wj,Ej)Gj=Vj,Wj,EjG_(j)=(V_(j),W_(j),E_(j))G_{j}=\left(\mathcal{V}_{j}, \mathcal{W}_{j}, \mathcal{E}_{j}\right)Gj=(Vj,Wj,Ej) be the bipartite graph at the jjjjj th iteration. Because we will use an unbounded number of iterations, it is important to ensure that Gj+1Gj+1G_(j+1)G_{j+1}Gj+1 has more edges than "what the qualitative parameters of GjGjG_(j)G_{j}Gj would typically predict." One way to assign meaning to this vague phrase is to use the edge density #Ej/(#Vj#Wj)#Ej/#Vj#Wj#E_(j)//(#V_(j)#W_(j))\# \mathscr{E}_{j} /\left(\# \mathcal{V}_{j} \# \mathcal{W}_{j}\right)#Ej/(#Vj#Wj). Actually, in our case, we
should use the weighted density
In a completely different direction, we can use the special GCD structure of our graphs to come up with another "measure of quality" of our new graph compared to the old one. Recall the integers aj+1aj+1a_(j+1)a_{j+1}aj+1 and bj+1bj+1b_(j+1)b_{j+1}bj+1. We then have
If all pairs (m,n)(m,n)(m,n)(m, n)(m,n) on the right-hand side of (3.8) were due to a fixed divisor of size >[Q/(Nt)]/gcd(aj,bj)>[Q/(Nt)]/gcdâ¡aj,bj> [Q//(Nt)]//gcd(a_(j),b_(j))>[Q /(N t)] / \operatorname{gcd}\left(a_{j}, b_{j}\right)>[Q/(Nt)]/gcdâ¡(aj,bj), then we would conclude that
Let us see a different argument for why this quantity might be a good choice, by studying the effect of each prime p∈{p1,…,pj}p∈p1,…,pjp in{p_(1),dots,p_(j)}p \in\left\{p_{1}, \ldots, p_{j}\right\}p∈{p1,…,pj} to the parameters Q/aj,Q/bjQ/aj,Q/bjQ//a_(j),Q//b_(j)Q / a_{j}, Q / b_{j}Q/aj,Q/bj and [Q/(Nt)]/gcd(aj,bj)[Q/(Nt)]/gcdâ¡aj,bj[Q//(Nt)]//gcd(a_(j),b_(j))[Q /(N t)] / \operatorname{gcd}\left(a_{j}, b_{j}\right)[Q/(Nt)]/gcdâ¡(aj,bj) that control the size of m,nm,nm,nm, nm,n, and gcd(m,n)gcdâ¡(m,n)gcd(m,n)\operatorname{gcd}(m, n)gcdâ¡(m,n), respectively, in (3.8):
Case 1: p∣ajp∣ajp∣a_(j)p \mid a_{j}p∣aj and p∣bjp∣bjp∣b_(j)p \mid b_{j}p∣bj. Then ppppp reduces the upper bounds on the size of both mmmmm and nnnnn by a factor 1/p1/p1//p1 / p1/p. On the other hand, it also reduces the lower bound on their GCD (that affects both mmmmm and nnnnn ) by 1/p1/p1//p1 / p1/p. Hence, we are in a balanced situation.
Case 2: p∤ajp∤ajp∤a_(j)p \nmid a_{j}p∤aj and p∤bjp∤bjp∤b_(j)p \nmid b_{j}p∤bj. In this case, ppppp affects no parameters.
Case 3: p∣ajp∣ajp∣a_(j)p \mid a_{j}p∣aj and p∤bjp∤bjp∤b_(j)p \nmid b_{j}p∤bj. Then ppppp reduces the upper bound on mmmmm by a factor 1/p1/p1//p1 / p1/p, but it does not affect the bound on nnnnn nor on gcd(m,n)gcdâ¡(m,n)gcd(m,n)\operatorname{gcd}(m, n)gcdâ¡(m,n). This is an advantageous situation, gaining us a factor of ppppp compared to what we had. Accordingly, λjλjlambda_(j)\lambda_{j}λj is multiplied by ppppp in this case. This gain allows us to afford a big loss of vertices when falling in this "asymmetric" case (a proportion of 1−O(1/p)1−O(1/p)1-O(1//p)1-O(1 / p)1−O(1/p) ).
Case 4: p∤ajp∤ajp∤a_(j)p \nmid a_{j}p∤aj and p∣bjp∣bjp∣b_(j)p \mid b_{j}p∣bj. Then we gain a factor of ppppp as in the previous case.
Iteratively increasing λjλjlambda_(j)\lambda_{j}λj would be adequate for showing (3.6), by mimicking the ErdÅ‘s-Vaaler argument from Section 3.3. Unfortunately, it is not possible to guarantee that λjλjlambda_(j)\lambda_{j}λj increases at each stage because it is not sensitive enough to the edge density, and so this proposal also fails. However, we will show that (a small variation of) the hybrid quantity
can be made to increase at each step, while keeping track of the sizes of the vertex sets. We call qjqjq_(j)q_{j}qj the quality of GjGjG_(j)G_{j}Gj.
Let us now discuss how we might carry out the "quality increment" strategy. Given VjVjV_(j)\mathcal{V}_{j}Vj and pj+1pj+1p_(j+1)p_{j+1}pj+1, we wish to set Vj+1=Vj(k)Vj+1=Vj(k)V_(j+1)=V_(j)^((k))\mathcal{V}_{j+1}=\mathcal{V}_{j}^{(k)}Vj+1=Vj(k) and Wj+1=Wj(ℓ)Wj+1=Wj(â„“)W_(j+1)=W_(j)^((â„“))\mathcal{W}_{j+1}=\mathcal{W}_{j}^{(\ell)}Wj+1=Wj(â„“) for some k,ℓ∈{0,1}k,ℓ∈{0,1}k,â„“in{0,1}k, \ell \in\{0,1\}k,ℓ∈{0,1}. Let us call Gj(k,ℓ)Gj(k,â„“)G_(j)^((k,â„“))G_{j}^{(k, \ell)}Gj(k,â„“) each of the four potential choices for Gj+1Gj+1G_(j+1)G_{j+1}Gj+1. For their quality qj(k,ℓ)qj(k,â„“)q_(j)^((k,â„“))q_{j}^{(k, \ell)}qj(k,â„“), we have:
where δj(k,ℓ)δj(k,â„“)delta_(j)^((k,â„“))\delta_{j}^{(k, \ell)}δj(k,â„“) is the edge density of Gj(k,ℓ),α=μ(Vj(1))/μ(Vj)Gj(k,â„“),α=μVj(1)/μVjG_(j)^((k,â„“)),alpha=mu(V_(j)^((1)))//mu(V_(j))G_{j}^{(k, \ell)}, \alpha=\mu\left(\mathcal{V}_{j}^{(1)}\right) / \mu\left(\mathcal{V}_{j}\right)Gj(k,â„“),α=μ(Vj(1))/μ(Vj) is the proportion of vertices in VjVjV_(j)\mathcal{V}_{j}Vj that are divisible by pj+1pj+1p_(j+1)p_{j+1}pj+1, and similarly β=μ(Wj(1))/μ(Wj)β=μWj(1)/μWjbeta=mu(W_(j)^((1)))//mu(W_(j))\beta=\mu\left(\mathcal{W}_{j}^{(1)}\right) / \mu\left(\mathcal{W}_{j}\right)β=μ(Wj(1))/μ(Wj). In addition, we have
so that if one of the δj(k,ℓ)δj(k,â„“)delta_(j)^((k,â„“))\delta_{j}^{(k, \ell)}δj(k,â„“), s is smaller than δδdelta\deltaδ, some other must be larger. Such an unbalanced situation is advantageous, so let us assume that δj(k,ℓ)∼δjδj(k,â„“)∼δjdelta_(j)^((k,â„“))∼delta_(j)\delta_{j}^{(k, \ell)} \sim \delta_{j}δj(k,â„“)∼δj for all k,ℓk,â„“k,â„“k, \ellk,â„“.
It is important to remark here that the factor (1−1/p)−2(1−1/p)−2(1-1//p)^(-2)(1-1 / p)^{-2}(1−1/p)−2 in the case (k,ℓ)=(1,1)(k,â„“)=(1,1)(k,â„“)=(1,1)(k, \ell)=(1,1)(k,â„“)=(1,1) is essential (the factors (1−1/p)−1(1−1/p)−1(1-1//p)^(-1)(1-1 / p)^{-1}(1−1/p)−1 in the asymmetric cases are less important as it turns out). Without this extra factor, we would not have been able to guarantee that the quality stays at least as big as qjqjq_(j)q_{j}qj. Crucially, this factor originates from the weights φ(v)/vφ(v)/vvarphi(v)//v\varphi(v) / vφ(v)/v of the vertices that are naturally built in the Duffin-Schaeffer conjecture and that dampen down contributions from integers with too many prime divisors.
We now discuss the formal details of our iterative algorithm. We must first set up some notation. We say that G=(V,W,E,P,a,b)G=(V,W,E,P,a,b)G=(V,W,E,P,a,b)G=(\mathcal{V}, \mathcal{W}, \mathscr{E}, \mathscr{P}, a, b)G=(V,W,E,P,a,b) is a square-free GCDGCDGCDG C DGCD graph if:
VVV\mathcal{V}V and WWW\mathcal{W}W are nonempty, finite sets of square-free integers;
(V,W,E)(V,W,E)(V,W,E)(\mathcal{V}, \mathcal{W}, \mathcal{E})(V,W,E) is a bipartite graph, meaning that E⊆V×WE⊆V×WEsubeVxxW\mathcal{E} \subseteq \mathcal{V} \times \mathcal{W}E⊆V×W;
PPP\mathcal{P}P is a finite set of primes, and aaaaa and bbbbb divide ∏p∈Ppâˆp∈P pprod_(p inP)p\prod_{p \in \mathcal{P}} pâˆp∈Pp;
a∣va∣va∣va \mid va∣v and b∣wb∣wb∣wb \mid wb∣w for all (v,w)∈V×W(v,w)∈V×W(v,w)inVxxW(v, w) \in \mathcal{V} \times \mathcal{W}(v,w)∈V×W;
if (v,w)∈E(v,w)∈E(v,w)inE(v, w) \in \mathcal{E}(v,w)∈E and p∈Pp∈Pp inPp \in \mathcal{P}p∈P, then p∣gcd(v,w)p∣gcdâ¡(v,w)p∣gcd(v,w)p \mid \operatorname{gcd}(v, w)p∣gcdâ¡(v,w) precisely when p∣gcd(a,b)p∣gcdâ¡(a,b)p∣gcd(a,b)p \mid \operatorname{gcd}(a, b)p∣gcdâ¡(a,b).
We shall refer to (P,a,b)(P,a,b)(P,a,b)(\mathcal{P}, a, b)(P,a,b) as the multiplicative data of GGGGG. Furthermore, we defined the edge density of GGGGG by δ(G):=μ(E)μ(V)μ(W)δ(G):=μ(E)μ(V)μ(W)delta(G):=(mu(E))/(mu(V)mu(W))\delta(G):=\frac{\mu(\mathcal{E})}{\mu(\mathcal{V}) \mu(\mathcal{W})}δ(G):=μ(E)μ(V)μ(W), and its quality by
In addition, we define the set of "remaining large primes" of GGGGG by
R(G):={p∉P:p>5100,p∣gcd(v,w) for some (v,w)∈E}R(G):=p∉P:p>5100,p∣gcdâ¡(v,w) for some (v,w)∈ER(G):={p!inP:p > 5^(100),p∣gcd(v,w)" for some "(v,w)inE}\mathcal{R}(G):=\left\{p \notin \mathcal{P}: p>5^{100}, p \mid \operatorname{gcd}(v, w) \text { for some }(v, w) \in \mathcal{E}\right\}R(G):={p∉P:p>5100,p∣gcdâ¡(v,w) for some (v,w)∈E}
Finally, if G′=(V′,W′,E′,P′,a′,b′)G′=V′,W′,E′,P′,a′,b′G^(')=(V^('),W^('),E^('),P^('),a^('),b^('))G^{\prime}=\left(\mathcal{V}^{\prime}, \mathcal{W}^{\prime}, \mathcal{E}^{\prime}, \mathcal{P}^{\prime}, a^{\prime}, b^{\prime}\right)G′=(V′,W′,E′,P′,a′,b′) is another square-free GCD graph, we call it a subgraph of GGGGG if V′⊆V,W′⊆W,E′⊆E,P′⊇P,∏p∣a′,p∈Pp=a,∏p∣b′,p∈Pp=bV′⊆V,W′⊆W,E′⊆E,P′⊇P,âˆp∣a′,p∈P p=a,âˆp∣b′,p∈P p=bV^(')subeV,W^(')subeW,E^(')subeE,P^(')supeP,prod_(p∣a^('),p inP)p=a,prod_(p∣b^('),p inP)p=b\mathcal{V}^{\prime} \subseteq \mathcal{V}, \mathcal{W}^{\prime} \subseteq \mathcal{W}, \mathcal{E}^{\prime} \subseteq \mathcal{E}, \mathcal{P}^{\prime} \supseteq \mathcal{P}, \prod_{p \mid a^{\prime}, p \in \mathcal{P}} p=a, \prod_{p \mid b^{\prime}, p \in \mathcal{P}} p=bV′⊆V,W′⊆W,E′⊆E,P′⊇P,âˆp∣a′,p∈Pp=a,âˆp∣b′,p∈Pp=b.
Lemma 3.6 (The quality increment argument). Let G=(V,W,E,P,a,b)G=(V,W,E,P,a,b)G=(V,W,E,P,a,b)G=(\mathcal{V}, \mathcal{W}, \mathcal{E}, \mathcal{P}, a, b)G=(V,W,E,P,a,b) be a square-free GCD graph, let p∈R(G)p∈R(G)p inR(G)p \in \mathcal{R}(G)p∈R(G), and let α=μ({v∈V:p∣v})μ(V)α=μ({v∈V:p∣v})μ(V)alpha=(mu({v inV:p∣v}))/(mu(V))\alpha=\frac{\mu(\{v \in \mathcal{V}: p \mid v\})}{\mu(\mathcal{V})}α=μ({v∈V:p∣v})μ(V) and β=μ({w∈W:p∣w})μ(W)β=μ({w∈W:p∣w})μ(W)beta=(mu({w inW:p∣w}))/(mu(W))\beta=\frac{\mu(\{w \in \mathcal{W}: p \mid w\})}{\mu(\mathcal{W})}β=μ({w∈W:p∣w})μ(W).
Let u=max{αβ,(1−α)(1−β)}u=max{αβ,(1−α)(1−β)}u=max{alpha beta,(1-alpha)(1-beta)}u=\max \{\alpha \beta,(1-\alpha)(1-\beta)\}u=max{αβ,(1−α)(1−β)}. Then
Now, if (3.11) is true with k=ℓk=â„“k=â„“k=\ellk=â„“, part (a) of the lemma follows immediately by (3.10) upon taking G′=Gk,kG′=Gk,kG^(')=G_(k,k)G^{\prime}=G_{k, k}G′=Gk,k. Assume then that (3.11) fails when k=ℓk=â„“k=â„“k=\ellk=â„“. We separate two cases.
Case 1: max{α,β}>512/pmax{α,β}>512/pmax{alpha,beta} > 5^(12)//p\max \{\alpha, \beta\}>5^{12} / pmax{α,β}>512/p. We know that (3.11) holds for some choice of k≠ℓk≠ℓk!=â„“k \neq \ellk≠ℓ. Suppose that k=1k=1k=1k=1k=1 and ℓ=0â„“=0â„“=0\ell=0â„“=0 for the sake of concreteness; the other case is similar. Then, (3.10) implies
(b) Let c=(1−p−3/2)−1c=1−p−3/2−1c=(1-p^(-3//2))^(-1)c=\left(1-p^{-3 / 2}\right)^{-1}c=(1−p−3/2)−1. Using (3.10), we get a quality increment by letting G′=Gk,ℓG′=Gk,â„“G^(')=G_(k,â„“)G^{\prime}=G_{k, \ell}G′=Gk,â„“ if one of the following inequalities holds:
Let α=1−A/pα=1−A/palpha=1-A//p\alpha=1-A / pα=1−A/p and β=1−B/pβ=1−B/pbeta=1-B//p\beta=1-B / pβ=1−B/p with A,B∈[0,512]A,B∈0,512A,B in[0,5^(12)]A, B \in\left[0,5^{12}\right]A,B∈[0,512]. It suffices to show that
Case 2: q(GJ1)<t30q(G0)qGJ1<t30qG0q(G_(J_(1))) < t^(30)q(G_(0))q\left(G_{J_{1}}\right)<t^{30} q\left(G_{0}\right)q(GJ1)<t30q(G0). In this case, we do not have such a big quality gain, so we need to use that Lt(v,w)>100Lt(v,w)>100L_(t)(v,w) > 100L_{t}(v, w)>100Lt(v,w)>100 for all (v,w)∈Bt(v,w)∈Bt(v,w)inB_(t)(v, w) \in \mathscr{B}_{t}(v,w)∈Bt. But we must be very careful because this condition might be dominated by the prime divisors of the fixed integers aaaaa and bbbbb we are constructing. Before we proceed, note that (3.15) implies that
Let R=R(J1)R=RJ1R=R(J_(1))\mathcal{R}=\mathcal{R}\left(J_{1}\right)R=R(J1) and let p∈Rp∈Rp inRp \in \mathcal{R}p∈R. By the construction of GJ1,pGJ1,pG_(J_(1)),pG_{J_{1}}, pGJ1,p divides a proportion >1−512/p>1−512/p> 1-5^(12)//p>1-5^{12} / p>1−512/p of the vertex sets VJ1VJ1V_(J_(1))\mathcal{V}_{J_{1}}VJ1 and WJ1WJ1W_(J_(1))\mathcal{W}_{J_{1}}WJ1. Therefore,
Next, we apply repeatedly Lemma 3.6 to create a sequence of distinct primes pJ1+1,pJ1+2,…∈RpJ1+1,pJ1+2,…∈Rp_(J_(1)+1),p_(J_(1)+2),dots inRp_{J_{1}+1}, p_{J_{1}+2}, \ldots \in \mathcal{R}pJ1+1,pJ1+2,…∈R and of GCD graphs Gj′=(Vj′,Wj′,Ej′,{p1,…,pJ1+j},aj′,bj′)Gj′=Vj′,Wj′,Ej′,p1,…,pJ1+j,aj′,bj′G_(j)^(')=(V_(j)^('),W_(j)^('),E_(j)^('),{p_(1),dots,p_(J_(1)+j)},a_(j)^('),b_(j)^('))G_{j}^{\prime}=\left(\mathcal{V}_{j}^{\prime}, \mathcal{W}_{j}^{\prime}, \mathcal{E}_{j}^{\prime},\left\{p_{1}, \ldots, p_{J_{1}+j}\right\}, a_{j}^{\prime}, b_{j}^{\prime}\right)Gj′=(Vj′,Wj′,Ej′,{p1,…,pJ1+j},aj′,bj′), j=1,…j=1,…j=1,dotsj=1, \ldotsj=1,…, with Gj′Gj′G_(j)^(')G_{j}^{\prime}Gj′ a subgraph of Gj−1′Gj−1′G_(j-1)^(')G_{j-1}^{\prime}Gj−1′. This process will terminate, say after KKKKK steps, and we will arrive at a GCD graph GK′GK′G_(K)^(')G_{K}^{\prime}GK′ with R(GK′)=∅RGK′=∅R(G_(K)^('))=O/\mathcal{R}\left(G_{K}^{\prime}\right)=\emptysetR(GK′)=∅. By construction, we have
In particular, EK′≠∅EK′≠∅E_(K)^(')!=O/\mathscr{E}_{K}^{\prime} \neq \emptysetEK′≠∅. It remains to give an upper bound on q(GK′)qGK′q(G_(K)^('))q\left(G_{K}^{\prime}\right)q(GK′).
by arguing as in the proof of (3.7). We may then insert this inequality into the definition of q(GK′)qGK′q(G_(K)^('))q\left(G_{K}^{\prime}\right)q(GK′) and conclude that q(GK′)≪e−t32t2N2qGK′≪e−t32t2N2q(G_(K)^('))≪e^(-t^(32))t^(2)N^(2)q\left(G_{K}^{\prime}\right) \ll e^{-t^{32}} t^{2} N^{2}q(GK′)≪e−t32t2N2. Together with (3.19) and (3.16), this completes the proof of (3.6), and thus of Theorem 3.5 in this last case as well.
ACKNOWLEDGMENTS
The author is grateful to James Maynard for his comments on a preliminary version of the paper.
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The Bloch-Kato conjecture, formulated in [11], relates the cohomology of global Galois representations to the special values of LLLLL-functions. We briefly recall a weak form of the conjecture, which will suffice for this survey. Let L/QpL/QpL//Q_(p)L / \mathbf{Q}_{p}L/Qp be a finite extension, let KKKKK be a number field, and let VVVVV be a representation of ΓK=Gal(K¯/K)ΓK=Galâ¡(K¯/K)Gamma_(K)=Gal( bar(K)//K)\Gamma_{K}=\operatorname{Gal}(\bar{K} / K)ΓK=Galâ¡(K¯/K) on a finite-dimensional LLLLL-vector space. We suppose VVVVV is unramified outside finitely many primes and de Rham at the primes above ppppp. Then we may attach to VVVVV the following two objects:
Its LLLLL-function, which is the formal Euler product
where vvvvv varies over (finite) primes of KKKKK, and Pv(V,X)∈L[X]Pv(V,X)∈L[X]P_(v)(V,X)in L[X]P_{v}(V, X) \in L[X]Pv(V,X)∈L[X] is a local Euler factor depending on the restriction of VVVVV to a decomposition group at vvvvv. It is conjectured that, for any choice of isomorphism L¯≅CL¯≅Cbar(L)~=C\bar{L} \cong \mathbf{C}L¯≅C, this product converges for ℜ(s)≫0ℜ(s)≫0ℜ(s)≫0\Re(s) \gg 0ℜ(s)≫0 and and has meromorphic continuation to all of CCC\mathbf{C}C.
Its Selmer group Hf1(K,V)Hf1(K,V)H_(f)^(1)(K,V)H_{\mathrm{f}}^{1}(K, V)Hf1(K,V), a certain (finite-dimensional) subspace of the Galois cohomology group H1(K,V)H1(K,V)H^(1)(K,V)H^{1}(K, V)H1(K,V) determined by local conditions at each prime, defined in [11][11][11][11][11].
The full conjecture as formulated in [11] also determines the leading term of L(V∗,s)LV∗,sL(V^(**),s)L\left(V^{*}, s\right)L(V∗,s) at s=1s=1s=1s=1s=1 up to a ppppp-adic unit, in terms of the cohomology of an integral lattice T⊂VT⊂VT sub VT \subset VT⊂V.
This conjecture includes as special cases a wide variety of well-known results and conjectures. For example, when VVVVV is the 1-dimensional trivial representation, the weak conjecture states that ζK(s)ζK(s)zeta_(K)(s)\zeta_{K}(s)ζK(s) has a simple pole at s=1s=1s=1s=1s=1; and the strong conjecture (for all ppppp at once) is equivalent to the analytic class number formula, relating the residue at this pole to the class group and unit group of KKKKK. If V=Tp(E)⊗QpV=Tp(E)⊗QpV=T_(p)(E)oxQ_(p)V=T_{p}(E) \otimes \mathbf{Q}_{p}V=Tp(E)⊗Qp, where EEEEE is an elliptic curve over KKKKK and Tp(E)Tp(E)T_(p)(E)T_{p}(E)Tp(E) is its Tate module, then L(V∗,s)LV∗,sL(V^(**),s)L\left(V^{*}, s\right)L(V∗,s) is the Hasse-Weil LLLLL-function L(E/K,s)L(E/K,s)L(E//K,s)L(E / K, s)L(E/K,s), and we recover the Birch and Swinnerton-Dyer conjecture for EEEEE over KKKKK.
Critical values
The LLLLL-function L(V∗,s)LV∗,sL(V^(**),s)L\left(V^{*}, s\right)L(V∗,s) is expected to satisfy a functional equation relating L(V,s)L(V,s)L(V,s)L(V, s)L(V,s) and L(V∗,1−s)LV∗,1−sL(V^(**),1-s)L\left(V^{*}, 1-s\right)L(V∗,1−s), after multiplying by a suitable product of ΓΓGamma\GammaΓ-functions L∞(V∗,s)L∞V∗,sL_(oo)(V^(**),s)L_{\infty}\left(V^{*}, s\right)L∞(V∗,s) (determined by the Hodge-Tate weights of VVVVV at ppppp and the action of complex conjugation). These ΓΓGamma\GammaΓ-factors may have poles at s=1s=1s=1s=1s=1, forcing L(V∗,1)LV∗,1L(V^(**),1)L\left(V^{*}, 1\right)L(V∗,1) to vanish.
The Bloch-Kato conjecture is closely related to the Iwasawa main conjecture, in which the finite-dimensional Selmer group Hf1(K,V)Hf1(K,V)H_(f)^(1)(K,V)H_{\mathrm{f}}^{1}(K, V)Hf1(K,V) is replaced by a finitely-generated module over an Iwasawa algebra. This connection with Iwasawa theory, together with the proof of the Iwasawa main conjecture in this context by Mazur and Wiles, plays an important role in Huber and Kings' proof [33] of the Bloch-Kato conjecture for 1-dimensional representations of ΓQΓQGamma_(Q)\Gamma_{\mathbf{Q}}ΓQ.
2. WHAT IS AN EULER SYSTEM?
For KKKKK a number field and VVVVV a ΓKΓKGamma_(K)\Gamma_{K}ΓK-representation as in Section 1, we have the notion of an Euler system for VVVVV, defined as follows. Let SSSSS be a finite set of places of KKKKK containing all infinite places, all primes above ppppp and all primes at which VVVVV is ramified.
We define RRR\mathcal{R}R to be the collection of integral ideals of KKKKK of the form m=a⋅bm=aâ‹…bm=a*b\mathfrak{m}=\mathfrak{a} \cdot \mathfrak{b}m=aâ‹…b, where a is a square-free product of primes of KKKKK not in SSSSS, and bbb\mathfrak{b}b divides p∞p∞p^(oo)p^{\infty}p∞. For each m∈Rm∈RminR\mathfrak{m} \in \mathcal{R}m∈R, let c[m]c[m]c[m]c[\mathfrak{m}]c[m] be the ray class field modulo mmm\mathfrak{m}m. Then an Euler system for (T,S)(T,S)(T,S)(T, S)(T,S) is a collection of classes
coresK[m]K[q](c[mq])={Pq(V∗(1),σq−1)⋅c[m] if q∉Sc[m] if q∣pcoresK[m]K[q]â¡(c[mq])=PqV∗(1),σq−1â‹…c[m] if q∉Sc[m] if q∣pcores_(K[m])^(K[q])(c[mq])={[P_(q)(V^(**)(1),sigma_(q)^(-1))*c[m]," if "q!in S],[c[m]," if "q∣p]:}\operatorname{cores}_{K[\mathfrak{m}]}^{K[\mathfrak{q}]}(c[\mathfrak{m} \mathfrak{q}])= \begin{cases}P_{\mathfrak{q}}\left(V^{*}(1), \sigma_{\mathfrak{q}}^{-1}\right) \cdot c[\mathfrak{m}] & \text { if } \mathfrak{q} \notin S \\ c[\mathfrak{m}] & \text { if } \mathfrak{q} \mid p\end{cases}coresK[m]K[q]â¡(c[mq])={Pq(V∗(1),σq−1)â‹…c[m] if q∉Sc[m] if q∣p
where cores denotes the Galois corestriction (or norm) map, and σqσqsigma_(q)\sigma_{\mathfrak{q}}σq is the image of Frob qq_(q){ }_{\mathrm{q}}q in Gal(K[m]/K)Galâ¡(K[m]/K)Gal(K[m]//K)\operatorname{Gal}(K[\mathfrak{m}] / K)Galâ¡(K[m]/K). By an Euler system for VVVVV, we mean an Euler system for some (T,S)(T,S)(T,S)(T, S)(T,S). (These general definitions are due to Kato, Perrin-Riou, and Rubin, building on earlier work of Kolyvagin; the standard reference is [56].)
The crucial application of Euler systems is the following: if an Euler system exists for VVVVV whose image in H1(K,V)H1(K,V)H^(1)(K,V)H^{1}(K, V)H1(K,V) is non-zero (and VVVVV satisfies some auxiliary technical hypotheses), then we obtain a bound for the so-called relaxed Selmer group 11^(1){ }^{1}1
The relaxed Selmer group differs from the Bloch-Kato Selmer group in that we impose no local conditions at ppppp. More generally, under additional assumptions on VVVVV and ccc\mathbf{c}c, we can obtain finer statements taking into account local conditions at ppppp, and hence control the dimension of the Bloch-Kato Selmer group itself.
Euler systems are hence extremely powerful tools for bounding Selmer groups, as long as we can understand whether the image of ccc\mathbf{c}c in H1(K,V)H1(K,V)H^(1)(K,V)H^{1}(K, V)H1(K,V) is non-vanishing. In order to
1 See [49] for this formulation. Theorem 2.2 .3 of [56] is an equivalent result, but expressed in terms of a Selmer group for V∗(1)V∗(1)V^(**)(1)V^{*}(1)V∗(1), which is related to that of VVVVV by Poitou-Tate duality.
use an Euler system to prove new cases of the Bloch-Kato conjecture, one needs to establish a so-called explicit reciprocity law, which is a criterion for the non-vanishing of the Euler system in terms of the value L(V∗,1)LV∗,1L(V^(**),1)L\left(V^{*}, 1\right)L(V∗,1).
Challenges. In order to use Euler system theory to approach the Bloch-Kato conjecture, and other related problems such as the Iwasawa main conjecture, there are two major challenges to be overcome:
(1) Can we construct "natural" examples of Euler systems (satisfying appropriate local conditions), for interesting global Galois representations VVVVV ?
(2) Can we prove reciprocity laws relating the images of these Euler systems in H1(K,V)H1(K,V)H^(1)(K,V)H^{1}(K, V)H1(K,V) to the values of LLLLL-functions?
This was carried out by Kato [35] for the Galois representations arising from modular forms; but Kato's approach to proving explicit reciprocity laws has turned out to be difficult to generalise. More recently, in a series of works with various co-authors beginning with [40] (building on earlier work of Bertolini-Darmon-Rotger [6]), we developed a general strategy for overcoming these challenges, for Galois representations arising from automorphic forms for a range of reductive groups. We will describe this strategy in the remainder of this article.
Variants. A related concept is that of an anticyclotomic Euler system, in which KKKKK is a CM field, and we replace the ray-class fields c[m]c[m]c[m]c[\mathfrak{m}]c[m] with ring class fields associated to ideals of the real subfield K+K+K^(+)K^{+}K+. These arise naturally when VVVVV is conjugate self-dual, i.e. Vc=V∗(1)Vc=V∗(1)V^(c)=V^(**)(1)V^{c}=V^{*}(1)Vc=V∗(1) where ccccc denotes complex conjugation. The most familiar example is Kolyvagin's Euler system of Heegner points [39]; for more recent examples, see, e.g. [12, 15, 25]. Many of the techniques explained here for constructing and studying (full) Euler systems are also applicable to anticyclotomic Euler systems, and we shall discuss examples of both below.
A rather more distant cousin is the concept of a bipartite Euler system, which arises naturally in the context of level-raising congruences; cf. [31] for a general account, and [43] for a dramatic recent application to the Bloch-Kato conjecture. These require a rather different set of techniques, and we shall not discuss them further here.
The 1-critical condition. We conjectured in [49] that, in order to construct Euler systems for VVVVV by geometric means (i.e. as the images of motivic cohomology classes), we need to impose a condition on VVVVV : it needs to be 1 -critical.
However, our intended applications involve the Bloch-Kato conjecture for critical values of LLLLL-functions; so we need to construct Euler systems for representations that are 0 critical, rather than 1-critical. So we shall construct Euler systems for these representations in two stages: firstly, we shall construct Euler systems for auxiliary 1-critical representations VVVVV, using motivic cohomology; secondly, we shall " ppppp-adically deform" our Euler systems, in order to pass from these 1 -critical VVVVV to others which are 0 -critical. This will be discussed in Section 4 below.
(2) G=ResF/QGL2G=ResF/Qâ¡GL2G=Res_(F//Q)GL_(2)G=\operatorname{Res}_{F / \mathbf{Q}} \mathrm{GL}_{2}G=ResF/Qâ¡GL2 for FFFFF real quadratic, as in [26,41][26,41][26,41][26,41][26,41];
(3) G=GSp4G=GSp4G=GSp_(4)G=\mathrm{GSp}_{4}G=GSp4, as in [47];
(4) G=GSp4×GL1GL2G=GSp4×GL1GL2G=GSp_(4)xx_(GL_(1))GL_(2)G=\mathrm{GSp}_{4} \times{ }_{\mathrm{GL}_{1}} \mathrm{GL}_{2}G=GSp4×GL1GL2, as in [32];
(5) G=GU(2,1)G=GU(2,1)G=GU(2,1)G=\mathrm{GU}(2,1)G=GU(2,1), as in [48][48][48][48][48];
Each of these groups is naturally equipped with a Shimura datum (G,X)(G,X)(G,X)(G, \mathcal{X})(G,X). In examples (1)-(4), the reflex field EEEEE is QQQ\mathbf{Q}Q; in (5) and (6), it is the imaginary quadratic field used to define the unitary group. (One can also retrospectively interpret Kato's construction [35] in these terms, taking G=GL2G=GL2G=GL_(2)G=\mathrm{GL}_{2}G=GL2; and similarly Kolyvagin's anticyclotomic Euler system [39], which is in effect the n=1n=1n=1n=1n=1 case of example (6).)
we can thus obtain classes in the Galois cohomology of ρπÏÏ€rho_(pi)\rho_{\pi}ÏÏ€ as the images of classes in the πf−Ï€f−pi_(f)-\pi_{\mathrm{f}}-Ï€f− eigenspace of Hetd+1(YG(K)E,Vλ)Hetd+1YG(K)E,VλH_(et)^(d+1)(Y_(G)(K)_(E),V_(lambda))H_{\mathrm{et}}^{d+1}\left(Y_{G}(K)_{E}, V_{\lambda}\right)Hetd+1(YG(K)E,Vλ). (For simplicity, we shall sketch the construction below assuming λ=0λ=0lambda=0\lambda=0λ=0, and refer to the original papers for the case of general coefficients.)
Motivic cohomology. In order to construct classes in Hetd+1(YG(K)E,Vλ)Hetd+1YG(K)E,VλH_(et)^(d+1)(Y_(G)(K)_(E),V_(lambda))H_{\mathrm{et}}^{d+1}\left(Y_{G}(K)_{E}, V_{\lambda}\right)Hetd+1(YG(K)E,Vλ), we shall use two other, related cohomology theories:
Motivic cohomology (see [3]), which takes values in QQQ\mathbf{Q}Q-vector spaces (or ZZZ\mathbf{Z}Z-lattices in them), and is closely related to algebraic KKKKK-theory and Chow groups;
Deligne-Beilinson (or absolute Hodge) cohomology (see [34]), which takes values in RRR\mathbf{R}R-vector spaces, and has a relatively straightforward presentation in terms of pairs (ω,σ)(ω,σ)(omega,sigma)(\omega, \sigma)(ω,σ), where ωωomega\omegaω is an algebraic differential form, and σσsigma\sigmaσ a real-analytic antiderivative of Re(ω)Reâ¡(ω)Re(omega)\operatorname{Re}(\omega)Reâ¡(ω).
Pushforward maps. If (H,y)↪(G,X)(H,y)↪(G,X)(H,y)↪(G,X)(H, y) \hookrightarrow(G, X)(H,y)↪(G,X) is the inclusion of a sub-Shimura datum (with the same reflex field EEEEE ), then we obtain finite morphisms of algebraic varieties over EEEEE,
where EEEEE is the reflex field of (H,y)(H,y)(H,y)(H, y)(H,y). More generally, for each g∈G(Af)g∈GAfg in G(A_(f))g \in G\left(\mathbf{A}_{\mathrm{f}}\right)g∈G(Af) we have a map
for any variety YYYYY. If YYYYY is a modular curve (i.e. a Shimura variety for GL2GL2GL_(2)\mathrm{GL}_{2}GL2 ), then we have a canonical family of units: if Y1(N)Y1(N)Y_(1)(N)Y_{1}(N)Y1(N) is the Shimura variety of level {(∗011)modN}∗011modN{([**],[0,1],[1])mod N}\left\{\left(\begin{array}{cc}* \\ 0 & 1 \\ 1\end{array}\right) \bmod N\right\}{(∗011)modN}, then we have the Siegel unit
denoted cg0,1/Ncg0,1/N_(c)g_(0,1//N){ }_{c} g_{0,1 / N}cg0,1/N in the notation of [35] (where ccccc is an auxiliary integer coprime to the level). Crucially, we have an explicit formula for the image of this class in Deligne-Beilinson cohomology; it is given by
where E2E2E_(2)E_{2}E2 is a weight 2 Eisenstein series, and E0an (s)E0an (s)E_(0)^("an ")(s)E_{0}^{\text {an }}(s)E0an (s) is a family of real-analytic Eisenstein series depending on a parameter s∈Cs∈Cs inCs \in \mathbf{C}s∈C. (See also [38] for analogues with coefficients, related to Eisenstein series of higher weights.)
Rankin-Eisenstein classes and Rankin-Selberg integrals. We shall consider the following general setting: we consider a Shimura datum (H,y)(H,y)(H,y)(H, y)(H,y) equipped with an embedding ι:(H,y)→(G,X)ι:(H,y)→(G,X)iota:(H,y)rarr(G,X)\iota:(H, y) \rightarrow(G, \mathcal{X})ι:(H,y)→(G,X), and also with a family of maps
for some level KH,1(N)KH,1(N)K_(H,1)(N)K_{H, 1}(N)KH,1(N), which we call Eisenstein classes for HHHHH.
Remark 3.1. One might hope for a broader range of "Eisenstein classes" in motivic cohomology, associated to Eisenstein series on other groups which are not just copies of GL2GL2GL_(2)\mathrm{GL}_{2}GL2 's. However, this question seems to be very difficult; see [21] for some results in this direction for symplectic groups. If we could construct motivic classes associated to Eisenstein series for the Siegel parabolic of GSp2nGSp2nGSp_(2n)\mathrm{GSp}_{2 n}GSp2n (rather than the Klingen parabolic as in [21]), or for the analogous parabolic subgroup in the unitary group U(n,n)U(n,n)U(n,n)\mathrm{U}(n, n)U(n,n), then it would open the way towards a far wider range of Euler system constructions.
By a motivic Rankin-Eisenstein class for (G,X)(G,X)(G,X)(G, \mathcal{X})(G,X) (with trivial coefficients), we shall mean a class of the form
To define a Rankin-Eisenstein class, we need to choose the group HHHHH, and the maps ι:H→Gι:H→Giota:H rarr G\iota: H \rightarrow Gι:H→G and ψ:H→(GL2)tψ:H→GL2tpsi:H rarr(GL_(2))^(t)\psi: H \rightarrow\left(\mathrm{GL}_{2}\right)^{t}ψ:H→(GL2)t. To guide us in choosing these, we shall use "RankinSelberg-type" integral formulas for LLLLL-functions of automorphic representations. There are a wide range of such formulas, relating automorphic LLLLL-functions to integrals of the form
where Ean (si)Ean siE^("an ")(s_(i))E^{\text {an }}\left(s_{i}\right)Ean (si) are real-analytic GL2GL2GL_(2)\mathrm{GL}_{2}GL2 Eisenstein series, and ϕÏ•phi\phiÏ• is a cuspform in the space of πÏ€pi\piÏ€. We call these period integrals. Typically, one expects such an integral to evaluate to a product of one or more copies of the LLLLL-function of πÏ€pi\piÏ€, evaluated at some linear combination of the parameters sisis_(i)s_{i}si. For instance, the Rankin-Selberg integral for GL2×GL2GL2×GL2GL_(2)xxGL_(2)\mathrm{GL}_{2} \times \mathrm{GL}_{2}GL2×GL2 is of this form, as is Novodvorsky's formula for the LLLLL-functions of GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4 and GSp4×GL2GSp4×GL2GSp_(4)xxGL_(2)\mathrm{GSp}_{4} \times \mathrm{GL}_{2}GSp4×GL2.
Using the explicit formula (3.1) relating Siegel units to Eisenstein series, one can often show that the Deligne-Beilinson realisations of Rankin-Eisenstein classes also lead to integrals of the form (3.2), for suitably chosen ϕÏ•phi\phiÏ• and sisis_(i)s_{i}si. When this applies, we can use it to relate our motivic Rankin-Eisenstein classes to special values of LLLLL-functions 22^(2){ }^{2}2 (as was carried out in Beilinson's original paper [3] for the LLLLL-functions of GL2GL2GL_(2)\mathrm{GL}_{2}GL2 and GL2×GL2GL2×GL2GL_(2)xxGL_(2)\mathrm{GL}_{2} \times \mathrm{GL}_{2}GL2×GL2; see, e.g. [36,42][36,42][36,42][36,42][36,42] for more recent examples).
This gives one a guide to constructing "interesting" Rankin-Eisenstein classes for a given (G,X)(G,X)(G,X)(G, \mathcal{X})(G,X) : one first searches for a Rankin-Selberg integral describing the relevant LLLLL function, and then attempts to breathe motivic life into this real-analytic formula, interpreting it as the Deligne-Beilinson realisation of a motivic Rankin-Eisenstein class. One should hence interpret Rankin-Eisenstein classes as "motivic avatars" of Rankin-Selberg integral formulae.
In the anti-cyclotomic (t=0)(t=0)(t=0)(t=0)(t=0) case, the period integral (3.2) is more mysterious; but there are still a number of results and conjectures predicting that these period integrals should be related to values of LLLLL-functions. For instance, the Gan-Gross-Prasad conjecture [22] gives such a relation in the important cases SO(n)↪SO(n)×SO(n+1)SO(n)↪SO(n)×SO(n+1)SO(n)↪SO(n)xxSO(n+1)\mathrm{SO}(n) \hookrightarrow \mathrm{SO}(n) \times \mathrm{SO}(n+1)SO(n)↪SO(n)×SO(n+1) and U(n)↪U(n)↪U(n)↪U(n) \hookrightarrowU(n)↪U(n)×U(n+1)U(n)×U(n+1)U(n)xx U(n+1)U(n) \times U(n+1)U(n)×U(n+1). This conjecture has recently been proved in the unitary case [9], although the orthogonal case is still open. We refer to Sakellaridis-Venkatesh [58] for conjectural generalisations to other pairs (G,H)(G,H)(G,H)(G, H)(G,H).
Example 3.2. In our examples (1)-(6) above, we choose HHHHH and ttttt as follows:
The integral formulae for LLLLL-functions underlying examples (1) and (2) are, respectively, the classical Rankin-Selberg integral and Asai's integral formula for quadratic Hilbert modular forms. Cases (3) and (4) are related to Novodvorsky's integral formula for GSp4×GL2GSp4×GL2GSp_(4)xxGL_(2)\mathrm{GSp}_{4} \times \mathrm{GL}_{2}GSp4×GL2LLLLL-functions (with an additional Eisenstein series on the GL2GL2GL_(2)\mathrm{GL}_{2}GL2 factor in the former case); and case (5) to an integral studied by Gelbart and Piatetski-Shapiro in [23]. Example (6) is related to conjectures of Getz-Wambach [24] on Friedberg-Jacquet periods for automorphic representations of unitary groups.
Rankin-Eisenstein classes and norm relations. In order to build Euler systems (either full or anticyclotomic) from Rankin-Eisenstein classes, we need the following conditions to hold:
("Open orbit" condition) The group HHHHH has an open orbit on the product
where BGBGB_(G)B_{G}BG is a Borel subgroup of GGGGG, and HHHHH acts on G/BGG/BGG//B_(G)G / B_{G}G/BG via ιιiota\iotaι, and on (P1)tP1t(P^(1))^(t)\left(\mathbf{P}^{1}\right)^{t}(P1)t via ψψpsi\psiψ.
("Small stabiliser" condition) For a point uuuuu in the open orbit, let SuSuS_(u)S_{u}Su be the subgroup of HHHHH which fixes uuuuu and acts trivially on the fibre at uuuuu of the tautological (Gm)tGmt(G_(m))^(t)\left(\mathbf{G}_{m}\right)^{t}(Gm)t-bundle over (P1)tP1t(P^(1))^(t)\left(\mathbf{P}^{1}\right)^{t}(P1)t. Then we require that the image of SuSuS_(u)S_{u}Su has small image in the maximal torus quotient of HHHHH.
The role of the "small stabiliser" condition is to allow us to construct classes over field extensions. Since the connected components of the Shimura variety YGYGY_(G)Y_{G}YG are defined over abelian extensions of EEEEE, and the Galois action on the component group is described by class field theory, we can modify the Rankin-Eisenstein classes to define elements in Hetd+1(YG(K)F,Zp(d+1+t2))Hetd+1YG(K)F,Zpd+1+t2H_(et)^(d+1)(Y_(G)(K)_(F),Z_(p)((d+1+t)/(2)))H_{\mathrm{et}}^{d+1}\left(Y_{G}(K)_{F}, \mathbf{Z}_{p}\left(\frac{d+1+t}{2}\right)\right)Hetd+1(YG(K)F,Zp(d+1+t2)) for a fixed level KKKKK and varying abelian extensions F/EF/EF//EF / EF/E. The class of abelian extensions that arise will depend on the image of SuSuS_(u)S_{u}Su in the maximal torus quotient; in the examples (1)-(5) above, since Su={1}Su={1}S_(u)={1}S_{u}=\{1\}Su={1} and the splitting field of the Galois action is the full maximal abelian extension of EEEEE, so we obtain classes over all ray class fields of EEEEE. On the other hand, in example (6) we obtain only the anticyclotomic extension (as one would expect, since t=0t=0t=0t=0t=0 in this case).
A much more subtle problem is that of horizontal norm relations, comparing classes over E[m]E[m]E[m]E[\mathfrak{m}]E[m] and E[mq]E[mq]E[mq]E[\mathfrak{m} \mathfrak{q}]E[mq] for auxiliary primes q∤mq∤mq∤m\mathfrak{q} \nmid \mathfrak{m}q∤m, with the Euler factors PqPqP_(q)P_{\mathfrak{q}}Pq appearing in the comparison. The strategy developed in [47] and refined in [48] is to use multiplicity-one results in smooth representation theory to reduce the norm relation to a purely local calculation with zeta-integrals, which can then be computed explicitly to give the Euler factor. These multiplicity-one results are themselves closely bound up with the open-orbit condition; see [57[57[57[57[57.
Remark 3.3. The open-orbit condition, together with the assumption that 2c+t=1+d2c+t=1+d2c+t=1+d2 c+t=1+d2c+t=1+d, amount to stating that the diagonal map (ι,ψ):(H,y)↪(G,X)×(GL2,H)t(ι,ψ):(H,y)↪(G,X)×GL2,Ht(iota,psi):(H,y)↪(G,X)xx(GL_(2),H)^(t)(\iota, \psi):(H, y) \hookrightarrow(G, \mathcal{X}) \times\left(\mathrm{GL}_{2}, \mathbb{H}\right)^{t}(ι,ψ):(H,y)↪(G,X)×(GL2,H)t is a special pair of Shimura data in the sense of [59, DEFINITION 3.1]. We can thus interpret the "small stabiliser" condition, at least for t=0t=0t=0t=0t=0, as a criterion for when the special cycles studied in [59] extend to norm-compatible families over field extensions.
4. DEFORMATION TO CRITICAL VALUES
Critical values. The above methods allow us to define Euler systems for the automorphic
Galois representations Vπ=ρπ(d+1+t2)VÏ€=ÏÏ€d+1+t2V_(pi)=rho_(pi)((d+1+t)/(2))V_{\pi}=\rho_{\pi}\left(\frac{d+1+t}{2}\right)VÏ€=ÏÏ€(d+1+t2), where πÏ€pi\piÏ€ is cohomological in weight 0 ; and there are generalisations to representations which are cohomological for a certain range of nonzero weights λλlambda\lambdaλ, determined by branching laws for the restriction of algebraic representations from G~=G×(GL2)tG~=G×GL2ttilde(G)=G xx(GL_(2))^(t)\tilde{G}=G \times\left(\mathrm{GL}_{2}\right)^{t}G~=G×(GL2)t to HHHHH. Let us write Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 for the set of weights λλlambda\lambdaλ which are accessible by these methods, for some specific choice of HHHHH and ψψpsi\psiψ; this is a convex polyhedron in the weight lattice of GGGGG, cut out by finitely many linear inequalities. In the examples (1)-(5), one checks that for any πÏ€pi\piÏ€ whose weight lies in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1, the representation VπVÏ€V_(pi)V_{\pi}VÏ€ is 1-critical, consistently with the conjectures of [49].
However, our goal is to prove the Bloch-Kato conjecture for critical LLLLL-values; so we are interested in those λλlambda\lambdaλ such that, for πÏ€pi\piÏ€ of weight λλlambda\lambdaλ, the representation VVVVV is 0 -critical, so L(π∨,1−t2)Lπ∨,1−t2L(pi^(vv),(1-t)/(2))L\left(\pi^{\vee}, \frac{1-t}{2}\right)L(π∨,1−t2) is a critical value. The set of such λλlambda\lambdaλ is a finite disjoint union of polyhedral regions; and we let Σ0Σ0Sigma_(0)\Sigma_{0}Σ0 be one of these regions, chosen to be adjacent to Σ1Σ1Sigma_(1)\Sigma_{1}Σ1. In order to approach the Bloch-Kato conjecture, we need to find a way of "deforming" our Euler systems from Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 to Σ0Σ0Sigma_(0)\Sigma_{0}Σ0.
Remark 4.2. A slightly different, but related, numerology applies for anticyclotomic Euler systems. In these cases, the relevant LLLLL-value is always critical, but it lies at the centre of the functional equation, so it may be forced to vanish for sign reasons. Since the local root numbers at the infinite places depend on λλlambda\lambdaλ, we have some ranges of weights where the root number is +1 (where we expect interesting central LLLLL-values) and others where it is -1 (where we expect anticyclotomic Euler systems). These play the roles of the 0 -critical and 1 -critical regions in the case of full Euler systems.
The Bertolini-Darmon-Prasanna strategy. Although the " 0 -critical" and " 1 -critical" weight ranges are disjoint, we can relate them together ppppp-adically, using a strategy introduced by Bertolini, Darmon and Prasanna in [5].
The weights λλlambda\lambdaλ of cohomological representations can naturally be seen as points of a ppppp-adic analytic space WWW\mathcal{W}W (parametrising characters T(Zp)→Cp×TZp→Cp×T(Z_(p))rarrC_(p)^(xx)T\left(\mathbf{Z}_{p}\right) \rightarrow \mathbf{C}_{p}^{\times}T(Zp)→Cp×, where TTTTT is a maximal torus in GGGGG ). This space is isomorphic to a finite union of nnnnn-dimensional open discs, where nnnnn is the rank of GGGGG. Crucially, both Σ0Σ0Sigma_(0)\Sigma_{0}Σ0 and Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 are Zariski-dense in WWW\mathcal{W}W.
Hida theory shows that there is a finite flat covering E→WE→WErarrW\mathcal{E} \rightarrow \mathcal{W}E→W, the ordinary eigenvariety of GGGGG, whose points above a dominant integral weight λλlambda\lambdaλ ("classical points") biject with automorphic representations πÏ€pi\piÏ€ of GGGGG which are cohomological of weight λλlambda\lambdaλ and ppppp-ordinary.
We thus have two separate families of objects, indexed by different sets of classical points on EEE\mathcal{E}E :
at points whose weights lie in Σ0Σ0Sigma_(0)\Sigma_{0}Σ0, we have the critical values of the complex LLLLL-function;
at points whose weights lie in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1, we have Euler systems arising from motivic cohomology.
Our first goal will be to "analytically continue" the Euler system classes from Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 into Σ0Σ0Sigma_(0)\Sigma_{0}Σ0.
This is not all that we require, however, since we also need a relation between the resulting Euler system for each 0 -critical VVVVV and the LLLLL-value L(V∗,1)LV∗,1L(V^(**),1)L\left(V^{*}, 1\right)L(V∗,1). Relations of this kind are known as explicit reciprocity laws, and they are the crown jewels of Euler system theory. Following a strategy initiated in [5] and further developed in [37], in order to prove explicit reciprocity laws, we shall use a second kind of ppppp-adic deformation: besides deforming Euler system classes from Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 to Σ0Σ0Sigma_(0)\Sigma_{0}Σ0, we shall also deform LLLLL-values from Σ0Σ0Sigma_(0)\Sigma_{0}Σ0 into Σ1Σ1Sigma_(1)\Sigma_{1}Σ1. The strategy consists of the following steps:
(i) We shall construct a function on the eigenvariety - an "analytic ppppp-adic LLLLL function" - whose values in Σ0Σ0Sigma_(0)\Sigma_{0}Σ0 are critical LLLLL-values (modified by appropriate periods and Euler factors).
(ii) Using the Perrin-Riou regulator map of ppppp-adic Hodge theory, we construct a second analytic function on the eigenvariety - a "motivic ppppp-adic LLLLL-function" - whose value at some cohomological πÏ€pi\piÏ€ measures the non-triviality of Euler system classes for πÏ€pi\piÏ€ locally at ppppp.
Note that the motivic ppppp-adic LLLLL-function has no a priori reason to be related to complex LLLLL-values; however, its values in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 are by definition related to the Euler system classes (which arise from motivic cohomology, hence the terminology).
(iii) We shall prove a " ppppp-adic regulator formula", showing that the values of the analytic ppppp-adic LLLLL-function in at points in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 are related to the localisations of the Euler system classes at ppppp.
(iv) Using the regulator formula of step (iii), we can deduce that the motivic and analytic ppppp-adic LLLLL-function coincide at points in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1. Since weights lying in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 are Zariski-dense in EEE\mathcal{E}E, this implies the two ppppp-adic LLLLL-functions coincide in Σ0Σ0Sigma_(0)\Sigma_{0}Σ0 as well. Since the values of the analytic ppppp-adic LLLLL-function in Σ0Σ0Sigma_(0)\Sigma_{0}Σ0 are complex LLLLL-values, we obtain the sought-for explicit reciprocity law.
At the time of writing, this strategy has only been fully carried out for the examples (1) and (3) in our list, and partially for (4). However, the remaining cases are being treated in ongoing work of members of our research groups; and we expect the strategy to extend to many other Euler systems (both full and anticyclotomic) besides these.
5. CONSTRUCTING ppppp-ADIC LLLLL-FUNCTIONS
Coherent cohomology
To construct the analytic ppppp-adic LLLLL-function, we shall use the integral formula (3.2). Previously, for weights in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1, we interpreted this integral as a cup-product in DeligneBeilinson cohomology. We shall now give a different cohomological interpretation of the same formula, for weights in the range Σ0Σ0Sigma_(0)\Sigma_{0}Σ0. Following a strategy introduced by Harris [28,29], we can choose the cusp-form ϕÏ•phi\phiÏ•, and the Eisenstein series, to be harmonic differential forms (with controlled growth at the boundary) representing Dolbeault cohomology classes valued in automorphic vector bundles. These can then be interpreted algebraically, via the comparison between Dolbeault cohomology and Zariski sheaf cohomology. The upshot is that
L(π∨,1−t2)Lπ∨,1−t2L(pi^(vv),(1-t)/(2))L\left(\pi^{\vee}, \frac{1-t}{2}\right)L(π∨,1−t2) can be related to a cup-product in the cohomology of coherent sheaves on a smooth toroidal compactification ShK(H,y)Σtor ShK(H,y)Σtor Sh_(K)(H,y)_(Sigma)^("tor ")\mathrm{Sh}_{K}(H, y){ }_{\Sigma}^{\text {tor }}ShK(H,y)Σtor of ShK(H,y)ShK(H,y)Sh_(K)(H,y)\mathrm{Sh}_{K}(H, y)ShK(H,y).
Interpolation
In order to construct a ppppp-adic LLLLL-function, we need to show that the cohomology classes appearing in our formula for the LLLLL-function interpolate in Hida-type ppppp-adic families, and that the cup-product of these families makes sense.
Until recently, there was a fundamental limitation in the available techniques: we could only interpolate cohomology classes corresponding to holomorphic automorphic forms (i.e. degree 0 coherent cohomology), or (via Serre duality) those in the top-degree cohomology, which correspond to anti-holomorphic forms. This is an obstacle for our intended applications, since the integral formulas relevant for Euler systems always involve coherent cohomology in degrees close to the middle of the possible range. (More precisely, the relevant degree is d+t−12d+t−12(d+t-1)/(2)\frac{d+t-1}{2}d+t−12, where ttttt is the number of Eisenstein series present, which is typically 0,1,20,1,20,1,20,1,20,1,2.) So unless ddddd is rather small, using holomorphic or anti-holomorphic classes will not work.
A slightly wider range of "product type" examples arises when (G,X)(G,X)(G,X)(G, \mathcal{X})(G,X) is a product of two Shimura data (G1,X1)×(G2,X2)G1,X1×G2,X2(G_(1),X_(1))xx(G_(2),X_(2))\left(G_{1}, X_{1}\right) \times\left(G_{2}, X_{2}\right)(G1,X1)×(G2,X2) of approximately equal dimension, with dim(X1)−dim(X2)=t−1dimâ¡X1−dimâ¡X2=t−1dim(X_(1))-dim(X_(2))=t-1\operatorname{dim}\left(\mathcal{X}_{1}\right)-\operatorname{dim}\left(\mathcal{X}_{2}\right)=t-1dimâ¡(X1)−dimâ¡(X2)=t−1; then we can build a class in the correct degree as the product of an anti-holomorphic form on X1X1X_(1)X_{1}X1 and a holomorphic one on X2X2X_(2)X_{2}X2, and the resulting cup-products can often be understood as Petersson-type scalar products in Hida theory. For instance, the Rankin-Selberg integral formula can be analysed in this way [30]. However, for G=GSp4G=GSp4G=GSp_(4)G=\mathrm{GSp}_{4}G=GSp4 (with t=2t=2t=2t=2t=2 and d=3d=3d=3d=3d=3 ), we need to work with a class in coherent H2H2H^(2)H^{2}H2, and these are not seen by orthodox Hida theory.
Higher Hida theory. A beautiful solution to this problem is provided by the "higher Hida theory" developed in [55]. Pilloni's work shows that degree 1 coherent cohomology for the GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4 Shimura variety interpolates in a "partial" Hida family, with one weight fixed and the other varying ppppp-adically.
At present higher Hida theory, in the above sense, is only available for a few specific groups, although these include many of the ones relevant for Euler systems: besides GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4, the group GU(2,1)GU(2,1)GU(2,1)\mathrm{GU}(2,1)GU(2,1) is treated in [53], and Hilbert modular groups in [27] (in both cases assuming GGGGG is locally split at p)p)p)p)p). In the GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4 and GU(2,1)GU(2,1)GU(2,1)\mathrm{GU}(2,1)GU(2,1) cases the results are also slightly weaker than one might ideally hope, since we only obtain families in which one component of the weight is fixed and the others vary (so the resulting ppppp-adic LLLLL-functions have one variable fewer than one would expect). However, we expect that these restrictions will be lifted in future work.
Remark 5.1. A related theory, higher Coleman theory, has been developed by Boxer and Pilloni in [13]. This theory also serves to interpolate higher-degree cohomology in families, with all components of the weight varying; and the theory applies to any Shimura variety of abelian type. However, unlike the higher Hida theory of [55], this theory only applies to cohomology classes satisfying an "overconvergence" condition. This rules out the 2-parameter GL2GL2GL_(2)\mathrm{GL}_{2}GL2 Eisenstein family which plays a prominent role in the constructions of [45], as this Eisenstein series is not overconvergent. It may be possible to work around this problem by combining the higher Coleman theory of [13] with the theory of families of nearly-overconvergent modular forms for GL2GL2GL_(2)\mathrm{GL}_{2}GL2 introduced by Andreatta-Iovita [1]; but the technical obstacles in carrying this out would be formidable.
6. P-ADIC REGULATORS
We now turn to step (iii) of the BDP strategy: relating values of the analytic ppppp-adic LLLLL-function in the range Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 to the localisations at ppppp of the Euler system classes.
Syntomic cohomology. For all but finitely many primes, the Shimura variety has a smooth integral model over ZpZpZ_(p)\mathbf{Z}_{p}Zp, and the motivic Rankin-Eisenstein classes can be lifted to the cohomology of this integral model. This allows us to study them via another cohomology theory, Besser's rigid syntomic cohomology [7]. This is a cohomology theory for smooth ZpZpZ_(p)\mathbf{Z}_{p}Zp-schemes yyyyy, which has two vital properties:
The Fontaine-Messing-Niziol map induces the Bloch-Kato exponential map on Galois cohomology; so, for a class in Het∗(Y,n)Het∗(Y,n)H_(et)^(**)(Y,n)H_{\mathrm{et}}^{*}(Y, n)Het∗(Y,n) in the image of motivic cohomol-
ogy of yyyyy, one can express its Bloch-Kato logarithm via cup-products in syntomic cohomology ("syntomic regulators").
Rigid syntomic cohomology and its variant, fp-cohomology, were defined by Besser as a generalisation of Coleman's theory of ppppp-adic integration. It is computed by an explicit complex of sheaves which is a ppppp-adic analogue of the realanalytic Deligne-Beilinson complex: sections of this complex are pairs (ω,σ)(ω,σ)(omega,sigma)(\omega, \sigma)(ω,σ), where ωωomega\omegaω is an algebraic differential form, and σσsigma\sigmaσ is an overconvergent rigidanalytic differential form such that dσ=(1−φ)ωdσ=(1−φ)ωd sigma=(1-varphi)omegad \sigma=(1-\varphi) \omegadσ=(1−φ)ω, where φφvarphi\varphiφ is a local lift of the Frobenius of the special fibre.
In a series of works, beginning with the breakthrough [16] by Darmon-Rotger (see also [6,10,38][6,10,38][6,10,38][6,10,38][6,10,38] ), rigid syntomic cohomology has been systematically exploited to compute the Bloch-Kato logarithms of Rankin-Eisenstein classes when GGGGG is a product of copies of GL2GL2GL_(2)\mathrm{GL}_{2}GL2, in terms of Petersson products of (non-classical) ppppp-adic modular forms. These can then be interpreted as values of ppppp-adic LLLLL-functions in a "1-critical" region Σ1Σ1Sigma_(1)\Sigma_{1}Σ1. All of these ppppp-adic LLLLL-functions are "product type" settings in the sense explained above, involving coherent cohomology in either top or bottom degree.
Remark 6.1. A key role in these constructions is played by an explicit formula for the image of the Siegel unit in the syntomic cohomology of the ordinary locus of the modular curve, which is the ppppp-adic counterpart of equation (3.1): it is represented by the pair
where E0(p)E0(p)E_(0)^((p))E_{0}^{(p)}E0(p) is a ppppp-adic Eisenstein series of weight 0 .
We can thus understand these syntomic regulator formulae as ppppp-adic counterparts of the integral formula (3.2), with the integral understood via Coleman's ppppp-adic integration theory, and the real-analytic Eisenstein class replaced by a ppppp-adic one.
However, syntomic cohomology of the whole Shimura variety is not well-suited to explicit computations, since there is generally no global lift of the Frobenius of the special fibre. The first major problem is hence to express the pairing in terms of the syntomic cohomology of certain open subschemes of the Shimura variety which do possess an explicit Frobenius lift. This requires some results on the Hecke eigenspaces appearing in the rigid cohomology of Newton strata of the special fibre, which are the ℓ=pâ„“=pâ„“=p\ell=pâ„“=p counterparts of the vanishing theorems proved by Caraiani-Scholze [14] for ℓâ„“â„“\ellâ„“-adic cohomology for ℓ≠pℓ≠pâ„“!=p\ell \neq pℓ≠p.
The second major problem is to establish a link between rigid syntomic cohomology and coherent cohomology, for varieties admitting a Frobenius lifting. We succeeded in proving such a relation via a new a spectral sequence (the so-called Poznań spectral sequence)
which is a syntomic analogue of the Hodge-de Rham spectral sequence: its E1E1E_(1)E_{1}E1 page is given by the mapping fibre of 1−φ1−φ1-varphi1-\varphi1−φ on coherent cohomology, and its abutment is rigid syntomic cohomology. In the case of the ordinary locus of the modular curve, where all coherent cohomology in positive degrees vanishes, this reduces to the description of a syntomic class as a pair of global sections (ω,σ)(ω,σ)(omega,sigma)(\omega, \sigma)(ω,σ) as described above.
Thanks to this new spectral sequence, we were able to express the syntomic regulator of our Rankin-Eisenstein class for GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4 as a pairing in coherent cohomology, which we could identify as a specialisation of the pairing in higher Hida theory defining the ppppp-adic LLLLL-function. We can hence identify the logarithm of the GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4 Euler system class with a noncritical value of a ppppp-adic LLLLL-function. This is the first example of a ppppp-adic regulator formula where the ppppp-adic LLLLL-function is not of product type. We expect this strategy to be applicable to all the other Euler systems mentioned in Section 3 above. Cases (2) and (6) are currently work in progress by Giada Grossi, and by Andrew Graham and Waqar Shah, respectively; and case (5) is being explored by some members of our research groups.
7. DEFORMATION TO CRITICAL VALUES
We can now proceed to the final step of the Bertolini-Darmon-Prasanna strategy: deforming from Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 to Σ0Σ0Sigma_(0)\Sigma_{0}Σ0.
A generalisation of Coleman and Perrin-Riou's theory of "big logarithm" maps (cf. [37]) also allows us to define a motivic ppppp-adic LLLLL-function Lmot Lmot L^("mot ")\mathscr{L}^{\text {mot }}Lmot associated to the bottom class in our family of Euler systems. Perrin-Riou's local reciprocity formula implies that Lmot Lmot L^("mot ")\mathscr{L}^{\text {mot }}Lmot has an interpolation property both in Σ0Σ0Sigma_(0)\Sigma_{0}Σ0 and in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1. For classical points πÏ€pi\piÏ€ whose weights lie in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1, the value of Lmot Lmot L^("mot ")\mathscr{L}^{\text {mot }}Lmot interpolates the Bloch-Kato logarithm of the geometrically-defined Euler system class for VπVÏ€V_(pi)V_{\pi}VÏ€. Much more subtly, if we evaluate Lmot Lmot L^("mot ")\mathscr{L}^{\text {mot }}Lmot at points πÏ€pi\piÏ€ whose weights lie in Σ0Σ0Sigma_(0)\Sigma_{0}Σ0, it computes the image under the dual-exponential map of the bottom class in the Euler system for VπVÏ€V_(pi)V_{\pi}VÏ€ which we have just defined using analytic continuation.
We would like to make the following argument: "the regulator formula shows that Lmot Lmot L^("mot ")\mathscr{L}^{\text {mot }}Lmot and the analytic ppppp-adic LLLLL-function LLL\mathscr{L}L agree at points in Σ1Σ1Sigma_(1)\Sigma_{1}Σ1, and these are Zariskidense; so L=Lmot L=Lmot L=L^("mot ")\mathscr{L}=\mathscr{L}^{\text {mot }}L=Lmot everywhere". This is essentially how we proved an explicit reciprocity law for GL2×GL2GL2×GL2GL_(2)xxGL_(2)\mathrm{GL}_{2} \times \mathrm{GL}_{2}GL2×GL2 in [37]. Unfortunately, there are two subtle technical hitches which occur in making this argument precise for GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4.
The second is that, while Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 is indeed Zariski-dense in the eigenvariety, the function LLL\mathscr{L}L is only defined on a lower-dimensional "slice" of the eigenvariety (on which the GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4 form has weight (r1,r2)r1,r2(r_(1),r_(2))\left(r_{1}, r_{2}\right)(r1,r2), with r1r1r_(1)r_{1}r1 varying and r2r2r_(2)r_{2}r2 fixed), and the intersection of each individual slice with Σ1Σ1Sigma_(1)\Sigma_{1}Σ1 is not Zariski-dense in the slice.
In [50], we circumvented these problems in a somewhat indirect way, by appealing to a second, independent construction of an analytic ppppp-adic LLLLL-function, defined using Shalika models for GL4GL4GL_(4)\mathrm{GL}_{4}GL4 [20]. As written this construction shares with [45] the shortcoming of requiring r2r2r_(2)r_{2}r2 to be fixed, but the methods of [44] can be applied in order to extend this construction by varying r2r2r_(2)r_{2}r2 as well. Using this we were able to
The lack of an Eichler-Shimura isomorphism in families - or, more precisely, of an isomorphism between the sheaves in which LmotLmotL^(mot)\mathscr{L}^{\mathrm{mot}}Lmot and the GL4pGL4pGL_(4)p\mathrm{GL}_{4} pGL4p-adic LLLLL-function take values - can be dealt with via the so-called "leading term argument". This proceeds as follows. There is clearly a meromorphic isomorphism between these sheaves which maps one ppppp-adic LLLLL-function to the other (since both are clearly non-zero). 33^(3){ }^{3}3 If this meromorphic isomorphism degenerates to zero at some "bad" 0 -critical πÏ€pi\piÏ€, then the bottom class in our Euler system for πÏ€pi\piÏ€ lies in the kernel of the Perrin-Riou regulator. However, this would also apply to all the classes c[m]c[m]c[m]c[m]c[m] in this Euler system, for all values of mmmmm. So we obtain an Euler system satisfying a very strong local condition at ppppp; and a result of Mazur-Rubin [52] shows that this condition is so strong that it forces the entire Euler system to be zero. Hence we can replace all of these classes by their derivatives in the weight direction, which amounts to renormalising the Eichler-Shimura map to reduce its order of vanishing by 1 .
Iterating this process, we eventually obtain a non-trivial Euler system for πÏ€pi\piÏ€; and if L(π∨,1−t2)≠0Lπ∨,1−t2≠0L(pi^(vv),(1-t)/(2))!=0L\left(\pi^{\vee}, \frac{1-t}{2}\right) \neq 0L(π∨,1−t2)≠0, the bottom class of this Euler system is non-zero. We can now deduce the vanishing of Hf1(Q,Vπ)Hf1Q,VÏ€H_(f)^(1)(Q,V_(pi))H_{\mathrm{f}}^{1}\left(\mathbf{Q}, V_{\pi}\right)Hf1(Q,VÏ€), where Vπ=ρπ(d+1+t2)VÏ€=ÏÏ€d+1+t2V_(pi)=rho_(pi)((d+1+t)/(2))V_{\pi}=\rho_{\pi}\left(\frac{d+1+t}{2}\right)VÏ€=ÏÏ€(d+1+t2), as predicted by the Bloch-Kato conjecture.
If AAAAA is a paramodular abelian surface over QQQ\mathbf{Q}Q which is ordinary at ppppp, and has analytic rank 0 , then we can use the above approach to prove the finiteness of A(Q)A(Q)A(Q)A(\mathbf{Q})A(Q) (as predicted by the Birch-Swinnerton-Dyer conjecture), and of the ppppp-part of the Tate-Shafarevich group, under the assumption that the GSp4GSp4GSp_(4)\mathrm{GSp}_{4}GSp4 eigenvariety be smooth at the point corresponding to AAAAA. This is work in progress.
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Department of Mathematics, University College London, London WC1E 6BT, UK, and ETH Zürich, Switzerland, s.zerbes@ucl.ac.uk
COUNTING PROBLEMS: CLASS GROUPS, PRIMES, AND NUMBER FIELDS
LILLIAN B. PIERCE
ABSTRACT
Each number field has an associated finite abelian group, the class group, that records certain properties of arithmetic within the ring of integers of the field. The class group is well studied, yet also still mysterious. A central conjecture of Brumer and Silverman states that for each prime ℓâ„“â„“\ellâ„“, every number field has the property that its class group has very few elements of order ℓâ„“â„“\ellâ„“, where "very few" is measured relative to the absolute discriminant of the field. This paper surveys recent progress toward this conjecture, and outlines its close connections to counting prime numbers, counting number fields of fixed discriminant, and counting number fields of bounded discriminant.
MATHEMATICS SUBJECT CLASSIFICATION 2020
Primary 11R29; Secondary 11R45, 11N05
KEYWORDS
Class groups, counting number fields, distribution of primes
1. HISTORICAL PRELUDE
In a 1640 letter to Mersenne, Fermat stated that an odd prime ppppp satisfies p=x2+y2p=x2+y2p=x^(2)+y^(2)p=x^{2}+y^{2}p=x2+y2 if and only if p≡1(mod4)p≡1(mod4)p-=1(mod4)p \equiv 1(\bmod 4)p≡1(mod4). Roughly 90 years later, Euler learned of Fermat's statement through correspondence with Goldbach, and by 1749, he worked out a proof. This fits into a bigger question, which Euler studied as well: for each n≥1n≥1n >= 1n \geq 1n≥1, which primes can be written as p=x2+ny2p=x2+ny2p=x^(2)+ny^(2)p=x^{2}+n y^{2}p=x2+ny2 ? Even more generally: which binary quadratic forms ax2+bxy+cy2ax2+bxy+cy2ax^(2)+bxy+cy^(2)a x^{2}+b x y+c y^{2}ax2+bxy+cy2 represent a given integer mmmmm ? This question also motivated work of Lagrange and Legendre, and then appeared in Gauss's celebrated 1801 work Disquisitiones Arithmeticae; see [26].
Gauss partitioned binary quadratic forms of discriminant D=b2−4acD=b2−4acD=b^(2)-4acD=b^{2}-4 a cD=b2−4ac into equivalence classes under SL2(Z)SL2(Z)SL_(2)(Z)\mathrm{SL}_{2}(\mathbb{Z})SL2(Z) changes of variable. (Here we will speak only of fundamental discriminants DDDDD; for notes on the original setting, see [84].) Gauss showed that for each DDDDD there are finitely many such classes (the cardinality is the class number, denoted h(D)h(D)h(D)h(D)h(D) ), and verified that the classes obey a group law (composition). Based on extensive computation, Gauss noticed that as D→−∞D→−∞D rarr-ooD \rightarrow-\inftyD→−∞, small class numbers stopped appearing, writing: "Nullum dubium esse videtur, quin series adscriptae revera abruptae sint...Demonstrationes autem rigorosae harum observationum perdifficiles esse videntur." ("It seems beyond doubt that the sequences written down do indeed break off... However, rigorous proofs of these observations appear to be most difficult" [43, P. 13].) As D→+∞D→+∞D rarr+ooD \rightarrow+\inftyD→+∞, a quite different behavior seemed to appear, leading to a conjecture that h(D)=1h(D)=1h(D)=1h(D)=1h(D)=1 for infinitely many D>0D>0D > 0D>0D>0.
It is hard to exaggerate the interest these two conjectures have generated. In the 1830s, Dirichlet proved a class number formula, relating the class number h(D)h(D)h(D)h(D)h(D) of a (fundamental) discriminant DDDDD to the value L(1,χ)L(1,χ)L(1,chi)L(1, \chi)L(1,χ) of an LLLLL-function associated to a real primitive character χχchi\chiχ modulo DDDDD. Consequently, throughout the 1900s, Gauss's questions were studied via the theory of the complex-variable functions L(s,χ)L(s,χ)L(s,chi)L(s, \chi)L(s,χ). A remarkable series of works by Hecke, Deuring, Mordell, and Heilbronn confirmed that for D<0D<0D < 0D<0D<0 the class number h(D)h(D)h(D)h(D)h(D) attains any value only finitely many times. How many times? Famously, the work of Heegner, Baker, and Stark proved that there are 9 (fundamental) discriminants D<0D<0D < 0D<0D<0 with class number 1. In full generality, Goldfeld showed an effective lower bound for h(D)h(D)h(D)h(D)h(D) when D<0D<0D < 0D<0D<0 would follow from a specific case of the Birch-Swinnerton-Dyer conjecture, which was then verified by Gross and Zagier; see [42]. Now, for each 1≤N≤1001≤N≤1001 <= N <= 1001 \leq N \leq 1001≤N≤100, one may find the number of discriminants D<0D<0D < 0D<0D<0 with h(D)=Nh(D)=Nh(D)=Nh(D)=Nh(D)=N in [93]. As for the other conjecture, that infinitely many (fundamental) discriminants D>0D>0D > 0D>0D>0 have class number 1 , this remains open, and very mysterious. These historical antecedents hint at the intertwined currents of "counting" and the analytic study of LLLLL-functions, which will also be present in the work we will survey.
We briefly mention another historical motivation for the study of class numbers, namely the failure of unique factorization. For example, in the ring Z[−5],21=3⋅7Z[−5],21=3â‹…7Z[sqrt(-5)],21=3*7\mathbb{Z}[\sqrt{-5}], 21=3 \cdot 7Z[−5],21=3â‹…7 but it also factors into irreducible, nonassociated factors as (1+2−5)(1−2−5)(1+2−5)(1−2−5)(1+2sqrt(-5))(1-2sqrt(-5))(1+2 \sqrt{-5})(1-2 \sqrt{-5})(1+2−5)(1−2−5). Here is a problem where the failure of unique factorization has an impact. Suppose one is searching for solutions x,y,z∈Nx,y,z∈Nx,y,z inNx, y, z \in \mathbb{N}x,y,z∈N to the equation xp+yp=zpxp+yp=zpx^(p)+y^(p)=z^(p)x^{p}+y^{p}=z^{p}xp+yp=zp for a prime p≥3p≥3p >= 3p \geq 3p≥3. If a nontrivial
solution (x,y,z)(x,y,z)(x,y,z)(x, y, z)(x,y,z) exists, then for ζpζpzeta_(p)\zeta_{p}ζp a ppppp th root of unity, we could write
y⋅y⋯y=(z−x)(z−ζpx)⋯(z−ζpp−1x)yâ‹…y⋯y=(z−x)z−ζpx⋯z−ζpp−1xy*y cdots y=(z-x)(z-zeta_(p)x)cdots(z-zeta_(p)^(p-1)x)y \cdot y \cdots y=(z-x)\left(z-\zeta_{p} x\right) \cdots\left(z-\zeta_{p}^{p-1} x\right)yâ‹…y⋯y=(z−x)(z−ζpx)⋯(z−ζpp−1x)
If Z[ζp]ZζpZ[zeta_(p)]\mathbb{Z}\left[\zeta_{p}\right]Z[ζp] possesses unique factorization, two such factorizations cannot exist, so (x,y,z)(x,y,z)(x,y,z)(x, y, z)(x,y,z) cannot exist-verifying Fermat's Last Theorem for this exponent ppppp. But to the disappointment of many, unique factorization fails in Z[ζp]ZζpZ[zeta_(p)]\mathbb{Z}\left[\zeta_{p}\right]Z[ζp] for infinitely many ppppp. As Neukirch writes, "Realizing the failure of unique factorization in general has led to one of the grand events in the history of number theory, the discovery of ideal theory by Eduard Kummer" [69, cH. I §3].
1.1. The class group
Let K/QK/QK//QK / \mathbb{Q}K/Q be a number field of degree nnnnn, with associated ring of integers OKOKO_(K)\mathcal{O}_{K}OK. Every proper integral ideal a⊂OKa⊂OKa subO_(K)a \subset \mathcal{O}_{K}a⊂OK factors into a product of prime ideals p1⋯pkp1⋯pkp_(1)cdotsp_(k)\mathfrak{p}_{1} \cdots \mathfrak{p}_{k}p1⋯pk in a unique way (salvaging the notion of unique factorization). Moreover, the fractional ideals of KKKKK form an abelian group JKJKJ_(K)J_{K}JK, the free abelian group on the set of nonzero prime ideals of OKOKO_(K)\mathcal{O}_{K}OK. In the case that every ideal in JKJKJ_(K)J_{K}JK belongs to the subgroup PKPKP_(K)P_{K}PK of principal ideals, OKOKO_(K)\mathcal{O}_{K}OK is a principal ideal domain, and unique factorization holds in OKOKO_(K)\mathcal{O}_{K}OK. But more typically, some "expansion" occurs when passing to ideals; the class group of KKKKK is defined to measure this.
The elements in ClKClKCl_(K)\mathrm{Cl}_{K}ClK are ideal classes, and the cardinality |ClK|ClK|Cl_(K)|\left|\mathrm{Cl}_{K}\right||ClK| is the class number. The quotient JK/PKJK/PKJ_(K)//P_(K)J_{K} / P_{K}JK/PK is trivial (so that every ideal is a principal ideal, and |ClK|=1ClK=1|Cl_(K)|=1\left|\mathrm{Cl}_{K}\right|=1|ClK|=1 ) precisely when unique factorization holds in OKOKO_(K)\mathcal{O}_{K}OK. (Thus the above strategy for Fermat's Last Theorem works for ppppp if |ClQ(ζp)|=1ClQζp=1|Cl_(Q(zeta_(p)))|=1\left|\mathrm{Cl}_{\mathbb{Q}\left(\zeta_{p}\right)}\right|=1|ClQ(ζp)|=1. In fact, Kummer showed that as long as the class number of Q(ζp)QζpQ(zeta_(p))\mathbb{Q}\left(\zeta_{p}\right)Q(ζp) is indivisible by ppppp, the argument can be salvaged; see [31]. Such a prime is called a regular prime. Here is an open question: are there infinitely many regular prime numbers?)
By a result of Minkowski in the geometry of numbers, every ideal class in ClKClKCl_(K)\mathrm{Cl}_{K}ClK contains an integral ideal bbb\mathfrak{b}b with norm ℜ(b)=(OK:b)ℜ(b)=OK:bℜ(b)=(O_(K):b)\Re(\mathfrak{b})=\left(\mathcal{O}_{K}: \mathfrak{b}\right)ℜ(b)=(OK:b) satisfying
where DK=|Disc(K/Q)|DK=|Discâ¡(K/Q)|D_(K)=|Disc(K//Q)|D_{K}=|\operatorname{Disc}(K / \mathbb{Q})|DK=|Discâ¡(K/Q)| and sssss counts the pairs of complex embeddings of KKKKK. As there are finitely many integral ideals of any given norm, Landau deduced (see [68, тнM. 4.4]):
In particular, the class group of a number field KKKKK is always a finite abelian group. (Throughout, A≪κBA≪κBA≪_(kappa)BA \ll_{\kappa} BA≪κB indicates that there exists a constant CκCκC_(kappa)C_{\kappa}Cκ such that |A|≤CκB|A|≤CκB|A| <= C_(kappa)B|A| \leq C_{\kappa} B|A|≤CκB.)
When K=Q(D)K=Q(D)K=Q(sqrtD)K=\mathbb{Q}(\sqrt{D})K=Q(D) is a quadratic field, this relates in a precise way to Gauss's construction of the class number for binary quadratic forms of discriminant DDDDD (see [8]). In modern terms, Gauss asked whether for each h∈Nh∈Nh inNh \in \mathbb{N}h∈N, there are finitely many imaginary quadratic fields KKKKK with |ClK|=hClK=h|Cl_(K)|=h\left|\mathrm{Cl}_{K}\right|=h|ClK|=h ? (Yes.) Are there infinitely many real quadratic fields KKKKK with |ClK|=1ClK=1|Cl_(K)|=1\left|\mathrm{Cl}_{K}\right|=1|ClK|=1 ? (We do not know.) In fact, here is an open question: are there infinitely many number fields, of arbitrary degrees, with class number 1? Here is another open question: are
there infinitely many number fields, of arbitrary degrees, with bounded class number? These difficult questions must consider the regulator RKRKR_(K)R_{K}RK of the field KKKKK, due to the (ineffective) inequalities by Siegel (for quadratic fields) and Brauer (in general) [68, Ñн. 8]:
DK1/2−ε≪n,ε|ClK|RK≪n,εDK1/2+ε, for all ε>0DK1/2−ε≪n,εClKRK≪n,εDK1/2+ε, for all ε>0D_(K)^(1//2-epsi)≪_(n,epsi)|Cl_(K)|R_(K)≪_(n,epsi)D_(K)^(1//2+epsi),quad" for all "epsi > 0D_{K}^{1 / 2-\varepsilon} \ll_{n, \varepsilon}\left|\mathrm{Cl}_{K}\right| R_{K} \ll_{n, \varepsilon} D_{K}^{1 / 2+\varepsilon}, \quad \text { for all } \varepsilon>0DK1/2−ε≪n,ε|ClK|RK≪n,εDK1/2+ε, for all ε>0
2. THE ℓâ„“â„“\ellâ„“-TORSION CONJECTURE
In addition to studying the size of the class group, it is also natural to study its structure. We will focus on the ℓâ„“â„“\ellâ„“-torsion subgroup, defined for each integer ℓ≥2ℓ≥2â„“ >= 2\ell \geq 2ℓ≥2 by
For example, the class number is divisible by a prime ℓâ„“â„“\ellâ„“ precisely when |ClK[ℓ]|>1ClK[â„“]>1|Cl_(K)[â„“]| > 1\left|\mathrm{Cl}_{K}[\ell]\right|>1|ClK[â„“]|>1. Related problems include studying the exponent of the class group, or counting how many number fields of a certain degree have class number divisible, or indivisible, by a given prime ℓâ„“â„“\ellâ„“. Such problems are addressed for imaginary quadratic fields in [4,44,45,82][4,44,45,82][4,44,45,82][4,44,45,82][4,44,45,82].
In this survey, we will focus on upper bounds for the ℓâ„“â„“\ellâ„“-torsion subgroup. The Minkowski bound (1.2) provides an upper bound for any field of degree nnnnn, and all ℓâ„“â„“\ellâ„“ :
(2.1)1≤|ClK[ℓ]|≤|ClK|≪n,εDK1/2+ε, for all ε>0(2.1)1≤ClK[â„“]≤ClK≪n,εDK1/2+ε, for all ε>0{:(2.1)1 <= |Cl_(K)[â„“]| <= |Cl_(K)|≪_(n,epsi)D_(K)^(1//2+epsi)","quad" for all "epsi > 0:}\begin{equation*}
1 \leq\left|\mathrm{Cl}_{K}[\ell]\right| \leq\left|\mathrm{Cl}_{K}\right| \ll_{n, \varepsilon} D_{K}^{1 / 2+\varepsilon}, \quad \text { for all } \varepsilon>0 \tag{2.1}
\end{equation*}(2.1)1≤|ClK[ℓ]|≤|ClK|≪n,εDK1/2+ε, for all ε>0
Our subject is a conjecture on the size of the ℓâ„“â„“\ellâ„“-torsion subgroup, which suggests that (2.1) is far from the truth. We will focus primarily on cases when ℓâ„“â„“\ellâ„“ is prime, since |ClK[m]|ClK[m]|Cl_(K)[m]|\left|\mathrm{Cl}_{K}[m]\right||ClK[m]| is multiplicative as a function of mmmmm, and for a prime ℓ,|ClK[ℓt]|≤|ClK[ℓ]|tâ„“,ClKâ„“t≤ClK[â„“]tâ„“,|Cl_(K)[â„“^(t)]| <= |Cl_(K)[â„“]|^(t)\ell,\left|\mathrm{Cl}_{K}\left[\ell^{t}\right]\right| \leq\left|\mathrm{Cl}_{K}[\ell]\right|^{t}â„“,|ClK[â„“t]|≤|ClK[â„“]|t (see [73]).
Conjecture 2.1 ( ℓâ„“â„“\ellâ„“-torsion conjecture). Fix a degree n≥2n≥2n >= 2n \geq 2n≥2 and a prime ℓâ„“â„“\ellâ„“. Every number field K/QK/QK//QK / \mathbb{Q}K/Q of degree nnnnn satisfies |ClK[ℓ]|≪n,ℓ,εDKεClK[â„“]≪n,â„“,εDKε|Cl_(K)[â„“]|≪_(n,â„“,epsi)D_(K)^(epsi)\left|\mathrm{Cl}_{K}[\ell]\right| \ll_{n, \ell, \varepsilon} D_{K}^{\varepsilon}|ClK[â„“]|≪n,â„“,εDKε for all ε>0ε>0epsi > 0\varepsilon>0ε>0.
This conjecture is due to Brumer and Silverman, in the more precise form: is it always true that logℓ|ClK[ℓ]|≪n,ℓlogDK/loglogDK[17logâ„“â¡ClK[â„“]≪n,â„“logâ¡DK/logâ¡logâ¡DK[17log_(â„“)|Cl_(K)[â„“]|≪_(n,â„“)log D_(K)//log log D_(K)[17\log _{\ell}\left|\mathrm{Cl}_{K}[\ell]\right| \ll_{n, \ell} \log D_{K} / \log \log D_{K}[17logâ„“â¡|ClK[â„“]|≪n,â„“logâ¡DK/logâ¡logâ¡DK[17, QUEstion Cl(ℓ,d)]Cl(â„“,d)]Cl(â„“,d)]\mathrm{Cl}(\ell, d)]Cl(â„“,d)] ? Brumer and Silverman were motivated by counting elliptic curves of fixed conductor. Subsequently, this conjecture has appeared in many further contexts, including bounding the ranks of elliptic curves [34, $1.2]; bounding Selmer groups and ranks of hyperelliptic curves [10]; counting number fields [29, P. 166]; studying equidistribution of CM points on Shimura varieties [98, CONJECTURE 3.5]; and counting nonuniform lattices in semisimple Lie groups [6, THM. 7.5].
Conjecture 2.1 is known to be true for the degree n=2n=2n=2n=2n=2 and the prime ℓ=2â„“=2â„“=2\ell=2â„“=2, when it follows from the genus theory of Gauss (see [68, Ñн. 8.3]). This is the only case in which it is known. Nevertheless, starting in the early 2000s, significant progress has been made. The purpose of this survey is to give some insight into the wide variety of methods developed in recent work toward the conjecture. As an initial measure of progress, we define:
Property Cn,ℓ(Δ)Cn,â„“(Δ)C_(n,â„“)(Delta)\mathbf{C}_{n, \ell}(\Delta)Cn,â„“(Δ). Fix a degree n≥2n≥2n >= 2n \geq 2n≥2 and a prime ℓâ„“â„“\ellâ„“. Property Cn,ℓ(Δ)Cn,â„“(Δ)C_(n,â„“)(Delta)\mathbf{C}_{n, \ell}(\Delta)Cn,â„“(Δ) holds if for all number fields K/QK/QK//QK / \mathbb{Q}K/Q of degree n,|ClK[ℓ]|≪n,ℓ,Δ,εDKΔ+εn,ClK[â„“]≪n,â„“,Δ,εDKΔ+εn,|Cl_(K)[â„“]|≪_(n,â„“,Delta,epsi)D_(K)^(Delta+epsi)n,\left|\mathrm{Cl}_{K}[\ell]\right| \ll_{n, \ell, \Delta, \varepsilon} D_{K}^{\Delta+\varepsilon}n,|ClK[â„“]|≪n,â„“,Δ,εDKΔ+ε for all ε>0ε>0epsi > 0\varepsilon>0ε>0.
Gauss proved that C2,2(0)C2,2(0)C_(2,2)(0)\mathbf{C}_{2,2}(0)C2,2(0) holds. Until recently, no other case with Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2 was known.
The first progress was for imaginary quadratic fields. Suppose K=Q(−d)K=Q(−d)K=Q(sqrt(-d))K=\mathbb{Q}(\sqrt{-d})K=Q(−d) for a square-free integer d>1d>1d > 1d>1d>1, and suppose that [a][a][a][a][a] is a nontrivial element in ClK[ℓ]ClK[â„“]Cl_(K)[â„“]\mathrm{Cl}_{K}[\ell]ClK[â„“] for a prime ℓ≥3ℓ≥3â„“ >= 3\ell \geq 3ℓ≥3; thus [a][a][a][a][a] is the principal ideal class. Then by the Minkowski bound (1.1), there exists an integral ideal bbb\mathfrak{b}b in [a][a][a][\mathfrak{a}][a] such that ℜ(b)≪d1/2ℜ(b)≪d1/2ℜ(b)≪d^(1//2)\Re(\mathfrak{b}) \ll d^{1 / 2}ℜ(b)≪d1/2. Moreover, bℓbâ„“b^(â„“)\mathfrak{b}^{\ell}bâ„“ is principal, say, generated by (y+z−d)/2(y+z−d)/2(y+zsqrt(-d))//2(y+z \sqrt{-d}) / 2(y+z−d)/2 for some integers y,zy,zy,zy, zy,z, and so (ℜ(b))ℓ=ℜ(bℓ)=(y2+dz2)/4(ℜ(b))â„“=ℜbâ„“=y2+dz2/4(ℜ(b))^(â„“)=ℜ(b^(â„“))=(y^(2)+dz^(2))//4(\Re(\mathfrak{b}))^{\ell}=\Re\left(\mathfrak{b}^{\ell}\right)=\left(y^{2}+d z^{2}\right) / 4(ℜ(b))â„“=ℜ(bâ„“)=(y2+dz2)/4. Consequently, |ClK[ℓ]|ClK[â„“]|Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right||ClK[â„“]| can be dominated (up to a factor dεdεd^(epsi)d^{\varepsilon}dε ) by the number of integral solutions to
(2.2)4xℓ=y2+dz2, with x≪d1/2,y≪dℓ/4,z≪dℓ/4−1/2. (2.2)4xâ„“=y2+dz2, with x≪d1/2,y≪dâ„“/4,z≪dâ„“/4−1/2. {:(2.2)4x^(â„“)=y^(2)+dz^(2)","quad" with "x≪d^(1//2)","y≪d^(â„“//4)","z≪d^(â„“//4-1//2)". ":}\begin{equation*}
4 x^{\ell}=y^{2}+d z^{2}, \quad \text { with } x \ll d^{1 / 2}, y \ll d^{\ell / 4}, z \ll d^{\ell / 4-1 / 2} \text {. } \tag{2.2}
\end{equation*}(2.2)4xâ„“=y2+dz2, with x≪d1/2,y≪dâ„“/4,z≪dâ„“/4−1/2.Â
When ℓ=3â„“=3â„“=3\ell=3â„“=3, this can be interpreted in several ways: counting solutions to a congruence y2=4x3(modd)y2=4x3(modd)y^(2)=4x^(3)(mod d)y^{2}=4 x^{3}(\bmod d)y2=4x3(modd); counting perfect square values of the polynomial f(x,z)=4x3−dz2f(x,z)=4x3−dz2f(x,z)=4x^(3)-dz^(2)f(x, z)=4 x^{3}-d z^{2}f(x,z)=4x3−dz2; or counting integral points on a family of Mordell elliptic curves y2=4x3−Dy2=4x3−Dy^(2)=4x^(3)-Dy^{2}=4 x^{3}-Dy2=4x3−D, with D=dz2D=dz2D=dz^(2)D=d z^{2}D=dz2. Pierce used the first two perspectives, and Helfgott and Venkatesh used the third perspective, to prove for the first time that property C2,3(Δ)C2,3(Δ)C_(2,3)(Delta)\mathbf{C}_{2,3}(\Delta)C2,3(Δ) holds for some Δ<1/2[48,70Δ<1/2[48,70Delta < 1//2[48,70\Delta<1 / 2[48,70Δ<1/2[48,70, 71]. (The Scholz reflection principle shows that log3|ClQ(−d)[3]|log3â¡ClQ(−d)[3]log_(3)|Cl_(Q(sqrt(-d)))[3]|\log _{3}\left|\mathrm{Cl}_{\mathbb{Q}(\sqrt{-d})}[3]\right|log3â¡|ClQ(−d)[3]| and log3|ClQ(3d)[3]|log3â¡ClQ(3d)[3]log_(3)|Cl_(Q(sqrt(3d)))[3]|\log _{3}\left|\mathrm{Cl}_{\mathbb{Q}(\sqrt{3 d})}[3]\right|log3â¡|ClQ(3d)[3]| differ by at most 1 , so results for 3-torsion apply comparably to both real and imaginary quadratic fields [76].) When ℓ≥5ℓ≥5â„“ >= 5\ell \geq 5ℓ≥5, the region in which x,y,zx,y,zx,y,zx, y, zx,y,z lie in (2.2) becomes inconveniently large relative to the trivial bound (2.1). Here is an open question: for a prime ℓ≥5ℓ≥5â„“ >= 5\ell \geq 5ℓ≥5, are there at most ≪dΔ≪dΔ≪d^(Delta)\ll d^{\Delta}≪dΔ integral solutions to (2.2), for some Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2 ?
Recently, Bhargava, Taniguchi, Thorne, Tsimerman, and Zhao made a breakthrough on property Cn,2(Δ)Cn,2(Δ)C_(n,2)(Delta)\mathbf{C}_{n, 2}(\Delta)Cn,2(Δ) for all n≥3n≥3n >= 3n \geq 3n≥3. Fix a prime ℓâ„“â„“\ellâ„“ and a number field KKKKK of degree nnnnn. Given any nontrivial ideal class [a]∈ClK[ℓ][a]∈ClK[â„“][a]inCl_(K)[â„“][\mathfrak{a}] \in \mathrm{Cl}_{K}[\ell][a]∈ClK[â„“], they show it contains an integral ideal bbb\mathfrak{b}b with bℓbâ„“b^(â„“)\mathfrak{b}^{\ell}bâ„“ a principal ideal generated by an element ββbeta\betaβ lying in a well-proportioned "box." By an ingenious geometry of numbers argument, they show the number of such generators ββbeta\betaβ in the box is ≪DKℓ/2−1/2≪DKâ„“/2−1/2≪D_(K)^(â„“//2-1//2)\ll D_{K}^{\ell / 2-1 / 2}≪DKâ„“/2−1/2. If ℓ≥3ℓ≥3â„“ >= 3\ell \geq 3ℓ≥3, this far exceeds the trivial bound (2.1), but if ℓ=2â„“=2â„“=2\ell=2â„“=2, it slightly improves it. The striking refinement comes by recalling that any ββbeta\betaβ of interest must also have |NK/Q(β)|=ℜ(bℓ)=(ℜ(b))ℓNK/Q(β)=ℜbâ„“=(ℜ(b))â„“|N_(K//Q)(beta)|=ℜ(b^(â„“))=(ℜ(b))^(â„“)\left|N_{K / \mathbb{Q}}(\beta)\right|=\Re\left(b^{\ell}\right)=(\Re(\mathfrak{b}))^{\ell}|NK/Q(β)|=ℜ(bâ„“)=(ℜ(b))â„“ be a perfect ℓâ„“â„“\ellâ„“ th power of an integer, say, yℓyâ„“y^(â„“)y^{\ell}yâ„“. For ℓ=2â„“=2â„“=2\ell=2â„“=2, they apply a celebrated result of Bombieri and Pila to count integral solutions (x,y)(x,y)(x,y)(x, y)(x,y) to the degree nnnnn equation NK/Q(β+x)=y2NK/Q(β+x)=y2N_(K//Q)(beta+x)=y^(2)N_{K / \mathbb{Q}}(\beta+x)=y^{2}NK/Q(β+x)=y2 [15]. This strategy proves that property Cn,2(1/2−1/2n)Cn,2(1/2−1/2n)C_(n,2)(1//2-1//2n)\mathbf{C}_{n, 2}(1 / 2-1 / 2 n)Cn,2(1/2−1/2n) holds for all degrees n≥3n≥3n >= 3n \geq 3n≥3. Further refinements for degrees 3,4 show C3,2(0.2785…)C3,2(0.2785…)C_(3,2)(0.2785 dots)\mathbf{C}_{3,2}(0.2785 \ldots)C3,2(0.2785…) and C4,2(0.2785…)C4,2(0.2785…)C_(4,2)(0.2785 dots)\mathbf{C}_{4,2}(0.2785 \ldots)C4,2(0.2785…) hold; see [10].
Only two further nontrivial cases of property Cn,ℓ(Δ)Cn,â„“(Δ)C_(n,â„“)(Delta)\mathbf{C}_{n, \ell}(\Delta)Cn,â„“(Δ) are known, and for these we introduce the Ellenberg-Venkatesh criterion.
2.1. The Ellenberg-Venkatesh criterion
An important criterion for bounding ℓâ„“â„“\ellâ„“-torsion in the class group of a number field KKKKK relies on counting small primes that are noninert in KKKKK. The germ of the idea, which has been credited independently to Soundararajan and Michel, goes as follows. Suppose, for example, that K=Q(−d)K=Q(−d)K=Q(sqrt(-d))K=\mathbb{Q}(\sqrt{-d})K=Q(−d) is an imaginary quadratic field with ddddd square-free, and ℓâ„“â„“\ellâ„“ is an odd prime. Let HHHHH denote ClK[ℓ]ClK[â„“]Cl_(K)[â„“]\mathrm{Cl}_{K}[\ell]ClK[â„“]. Then |H|=|ClK|/[ClK:H]|H|=ClK/ClK:H|H|=|Cl_(K)|//[Cl_(K):H]|H|=\left|\mathrm{Cl}_{K}\right| /\left[\mathrm{Cl}_{K}: H\right]|H|=|ClK|/[ClK:H], and to show that |H||H||H||H||H| is small, it suffices to show that the index [ClK:H]ClK:H[Cl_(K):H]\left[\mathrm{Cl}_{K}: H\right][ClK:H] is large. Now suppose that p1≠p2p1≠p2p_(1)!=p_(2)p_{1} \neq p_{2}p1≠p2 are rational primes not dividing 2d2d2d2 d2d that both split in KKKKK, say, p1=p1p1σp1=p1p1σp_(1)=p_(1)p_(1)^(sigma)p_{1}=\mathfrak{p}_{1} \mathfrak{p}_{1}^{\sigma}p1=p1p1σ and p2=p2p2σp2=p2p2σp_(2)=p_(2)p_(2)^(sigma)p_{2}=\mathfrak{p}_{2} \mathfrak{p}_{2}^{\sigma}p2=p2p2σ,
where σσsigma\sigmaσ is the nontrivial automorphism of KKKKK. We claim that as long as p1,p2p1,p2p_(1),p_(2)p_{1}, p_{2}p1,p2 are sufficiently small, p1p1p_(1)\mathfrak{p}_{1}p1 and p2p2p_(2)\mathfrak{p}_{2}p2 must represent different cosets of HHHHH. Indeed, supposing to the contrary that p1H=p2Hp1H=p2Hp_(1)H=p_(2)H\mathfrak{p}_{1} H=\mathfrak{p}_{2} Hp1H=p2H, one deduces that p1p2σ∈Hp1p2σ∈Hp_(1)p_(2)^(sigma)in H\mathfrak{p}_{1} \mathfrak{p}_{2}^{\sigma} \in Hp1p2σ∈H so that (p1p2σ)ℓp1p2σℓ(p_(1)p_(2)^(sigma))^(â„“)\left(\mathfrak{p}_{1} \mathfrak{p}_{2}^{\sigma}\right)^{\ell}(p1p2σ)â„“ is a principal ideal, say, generated by (y+z−d)/2(y+z−d)/2(y+zsqrt(-d))//2(y+z \sqrt{-d}) / 2(y+z−d)/2, for some y,z∈Zy,z∈Zy,z inZy, z \in \mathbb{Z}y,z∈Z. Taking norms shows
If p1,p2<(1/4)d1/(2ℓ)p1,p2<(1/4)d1/(2â„“)p_(1),p_(2) < (1//4)d^(1//(2â„“))p_{1}, p_{2}<(1 / 4) d^{1 /(2 \ell)}p1,p2<(1/4)d1/(2â„“), this forces z=0z=0z=0z=0z=0, which yields a contradiction, since 4(p1p2)ℓ4p1p2â„“4(p_(1)p_(2))^(â„“)4\left(p_{1} p_{2}\right)^{\ell}4(p1p2)â„“ cannot be a perfect square. This proves the claim. In particular, if there are MMMMM such distinct primes p1,…,pM<(1/4)d1/2ℓp1,…,pM<(1/4)d1/2â„“p_(1),dots,p_(M) < (1//4)d^(1//2â„“)p_{1}, \ldots, p_{M}<(1 / 4) d^{1 / 2 \ell}p1,…,pM<(1/4)d1/2â„“ with pj∤2dpj∤2dp_(j)∤2dp_{j} \nmid 2 dpj∤2d and pjpjp_(j)p_{j}pj split in KKKKK, then |ClK[ℓ]|≤|ClK|M−1ClK[â„“]≤ClKM−1|Cl_(K)[â„“]| <= |Cl_(K)|M^(-1)\left|\mathrm{Cl}_{K}[\ell]\right| \leq\left|\mathrm{Cl}_{K}\right| M^{-1}|ClK[â„“]|≤|ClK|M−1.
Ellenberg and Venkatesh significantly generalized this strategy to prove an influential criterion, which we state in the case of extensions of QQQ\mathbb{Q}Q [34]. (Throughout this survey, we will focus for simplicity on extensions of QQQ\mathbb{Q}Q, but many of the theorems and questions we mention have analogues in the literature over any fixed number field.)
Ellenberg-Venkatesh criterion. Suppose K/QK/QK//QK / \mathbb{Q}K/Q is a number field of degree n≥2n≥2n >= 2n \geq 2n≥2, fix an integer ℓ≥2ℓ≥2â„“ >= 2\ell \geq 2ℓ≥2, and fix η<12ℓ(n−1)η<12â„“(n−1)eta < (1)/(2â„“(n-1))\eta<\frac{1}{2 \ell(n-1)}η<12â„“(n−1). Suppose that there are MMMMM prime ideals p1,…,pM⊂OKp1,…,pM⊂OKp_(1),dots,p_(M)subO_(K)\mathfrak{p}_{1}, \ldots, \mathfrak{p}_{M} \subset \mathcal{O}_{K}p1,…,pM⊂OK such that each pjpjp_(j)\mathfrak{p}_{j}pj has norm ℜ(pj)<DKηℜpj<DKηℜ(p_(j)) < D_(K)^(eta)\mathfrak{\Re}\left(\mathfrak{p}_{j}\right)<D_{K}^{\eta}ℜ(pj)<DKη, pjpjp_(j)\mathfrak{p}_{j}pj is unramified in KKKKK and pjpjp_(j)\mathfrak{p}_{j}pj is not an extension of a prime ideal from any proper subfield of KKKKK. Then
(2.4)|ClK[ℓ]|≪n,ℓ,εDK12+εM−1, for all ε>0(2.4)ClK[â„“]≪n,â„“,εDK12+εM−1, for all ε>0{:(2.4)|Cl_(K)[â„“]|≪_(n,â„“,epsi)D_(K)^((1)/(2)+epsi)M^(-1)","quad" for all "epsi > 0:}\begin{equation*}
\left|\mathrm{Cl}_{K}[\ell]\right| \ll_{n, \ell, \varepsilon} D_{K}^{\frac{1}{2}+\varepsilon} M^{-1}, \quad \text { for all } \varepsilon>0 \tag{2.4}
\end{equation*}(2.4)|ClK[ℓ]|≪n,ℓ,εDK12+εM−1, for all ε>0
(A prime ideal p⊂OKp⊂OKpsubO_(K)\mathfrak{p} \subset \mathcal{O}_{K}p⊂OK lying above a prime p∈Qp∈Qp inQp \in \mathbb{Q}p∈Q is unramified in K/QK/QK//QK / \mathbb{Q}K/Q if p2∤pOKp2∤pOKp^(2)∤pO_(K)\mathfrak{p}^{2} \nmid p \mathcal{O}_{K}p2∤pOK; a prime ideal p⊂OKp⊂OKpsubO_(K)\mathfrak{p} \subset \mathcal{O}_{K}p⊂OK is an extension of a prime ideal in a proper subfield K0⊂KK0⊂KK_(0)sub KK_{0} \subset KK0⊂K if there exists a prime ideal p0⊂OK0p0⊂OK0p_(0)subO_(K_(0))\mathfrak{p}_{0} \subset \mathcal{O}_{K_{0}}p0⊂OK0 such that p=p0OKp=p0OKp=p_(0)O_(K)\mathfrak{p}=\mathfrak{p}_{0} \mathcal{O}_{K}p=p0OK.) For example, if p<DKηp<DKηp < D_(K)^(eta)p<D_{K}^{\eta}p<DKη is a rational prime that splits completely in KKKKK, so that pOK=p1⋯pnpOK=p1⋯pnpO_(K)=p_(1)cdotsp_(n)p \mathcal{O}_{K}=\mathfrak{p}_{1} \cdots \mathfrak{p}_{n}pOK=p1⋯pn for distinct prime ideals pjpjp_(j)\mathfrak{p}_{j}pj, then each pjpjp_(j)\mathfrak{p}_{j}pj satisfies the hypotheses of the criterion. In particular, if MMMMM rational (unramified) primes p1,…,pM<DKηp1,…,pM<DKηp_(1),dots,p_(M) < D_(K)^(eta)p_{1}, \ldots, p_{M}<D_{K}^{\eta}p1,…,pM<DKη split completely in KKKKK, then (2.4) holds. Alternatively, it suffices to exhibit prime ideals pj⊂OKpj⊂OKp_(j)subO_(K)\mathfrak{p}_{j} \subset \mathcal{O}_{K}pj⊂OK of degree 1 , since such a prime ideal cannot be an extension of a prime ideal from a proper subfield.
Here is one of Ellenberg and Venkatesh's striking applications, which shows that C2,3(1/3)C2,3(1/3)C_(2,3)(1//3)\mathbf{C}_{2,3}(1 / 3)C2,3(1/3) holds-the current record for n=2,ℓ=3n=2,â„“=3n=2,â„“=3n=2, \ell=3n=2,â„“=3. Fix a large square-free integer d>1d>1d > 1d>1d>1. Any prime p∤6dp∤6dp∤6dp \nmid 6 dp∤6d that is inert in Q(−3)Q(−3)Q(sqrt(-3))\mathbb{Q}(\sqrt{-3})Q(−3) must split either in Q(d)Q(d)Q(sqrtd)\mathbb{Q}(\sqrt{d})Q(d) or in Q(−3d)Q(−3d)Q(sqrt(-3d))\mathbb{Q}(\sqrt{-3 d})Q(−3d). Thus for any η<1/6η<1/6eta < 1//6\eta<1 / 6η<1/6, at least one field K∈{Q(d),Q(−3d)}K∈{Q(d),Q(−3d)}K in{Q(sqrtd),Q(sqrt(-3d))}K \in\{\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{-3 d})\}K∈{Q(d),Q(−3d)} has a positive proportion of the primes (1/2) dη≤p≤dηdη≤p≤dηd^(eta) <= p <= d^(eta)d^{\eta} \leq p \leq d^{\eta}dη≤p≤dη split in KKKKK. By the Ellenberg-Venkatesh criterion (2.4), this field KKKKK then has the property that |ClK[3]|≪DK1/3+εClK[3]≪DK1/3+ε|Cl_(K)[3]|≪D_(K)^(1//3+epsi)\left|\mathrm{Cl}_{K}[3]\right| \ll D_{K}^{1 / 3+\varepsilon}|ClK[3]|≪DK1/3+ε for all ε>0ε>0epsi > 0\varepsilon>0ε>0. By the Scholz reflection principle, this bound also applies to the other field in the pair, and C2,3(1/3)C2,3(1/3)C_(2,3)(1//3)\mathbf{C}_{2,3}(1 / 3)C2,3(1/3) holds.
The Scholz reflection principle has also been generalized by Ellenberg and Venkatesh to bound ℓâ„“â„“\ellâ„“-torsion (for odd primes ℓâ„“â„“\ellâ„“ ) in class groups of even-degree extensions of certain number fields. In particular, by pairing their criterion with a reflection principle, they show that C3,3(1/3)C3,3(1/3)C_(3,3)(1//3)\mathbf{C}_{3,3}(1 / 3)C3,3(1/3) holds and C4,3(Δ)C4,3(Δ)C_(4,3)(Delta)\mathbf{C}_{4,3}(\Delta)C4,3(Δ) holds for some Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2 [34, coR. 3.7]. This concludes the list of degrees nnnnn and primes ℓâ„“â„“\ellâ„“ for which property Cn,ℓ(Δ)Cn,â„“(Δ)C_(n,â„“)(Delta)\mathbf{C}_{n, \ell}(\Delta)Cn,â„“(Δ) is known for some Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2.
Here are open problems: reduce the value Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2 for which Cn,ℓ(Δ)Cn,â„“(Δ)C_(n,â„“)(Delta)\mathbf{C}_{n, \ell}(\Delta)Cn,â„“(Δ) holds, when n≥3n≥3n >= 3n \geq 3n≥3 and ℓ=2â„“=2â„“=2\ell=2â„“=2, or when n=2,3n=2,3n=2,3n=2,3n=2,3 or 4 and ℓ=3â„“=3â„“=3\ell=3â„“=3. For n=2,3n=2,3n=2,3n=2,3n=2,3 or 4 and a prime ℓ≥5ℓ≥5â„“ >= 5\ell \geq 5ℓ≥5,
prove for the first time that Cn,ℓ(Δ)Cn,â„“(Δ)C_(n,â„“)(Delta)\mathbf{C}_{n, \ell}(\Delta)Cn,â„“(Δ) holds for some Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2. For n≥5n≥5n >= 5n \geq 5n≥5 and a prime ℓ≥3ℓ≥3â„“ >= 3\ell \geq 3ℓ≥3, prove for the first time that Cn,ℓ(Δ)Cn,â„“(Δ)C_(n,â„“)(Delta)\mathbf{C}_{n, \ell}(\Delta)Cn,â„“(Δ) holds for some Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2.
The Ellenberg-Venkatesh criterion underlies most of the significant recent progress on bounding ℓâ„“â„“\ellâ„“-torsion in class groups. What is the best result it can imply? Assuming the Generalized Riemann Hypothesis, given any number field K/QK/QK//QK / \mathbb{Q}K/Q of degree nnnnn with DKDKD_(K)D_{K}DK sufficiently large, a positive proportion of primes p<DKηp<DKηp < D_(K)^(eta)p<D_{K}^{\eta}p<DKη split completely in KKKKK, implying
(2.5)|ClK[ℓ]|≪n,ℓ,εDK12−12ℓ(n−1)+ε, for all ε>0(2.5)ClK[â„“]≪n,â„“,εDK12−12â„“(n−1)+ε, for all ε>0{:(2.5)|Cl_(K)[â„“]|≪_(n,â„“,epsi)D_(K)^((1)/(2)-(1)/(2â„“(n-1))+epsi)","quad" for all "epsi > 0:}\begin{equation*}
\left|\mathrm{Cl}_{K}[\ell]\right| \ll_{n, \ell, \varepsilon} D_{K}^{\frac{1}{2}-\frac{1}{2 \ell(n-1)}+\varepsilon}, \quad \text { for all } \varepsilon>0 \tag{2.5}
\end{equation*}(2.5)|ClK[ℓ]|≪n,ℓ,εDK12−12ℓ(n−1)+ε, for all ε>0
As this is a useful benchmark, we will call this the GRH-bound, and for convenience set ΔGRH=12−12ℓ(n−1)ΔGRH=12−12â„“(n−1)Delta_(GRH)=(1)/(2)-(1)/(2â„“(n-1))\Delta_{\mathrm{GRH}}=\frac{1}{2}-\frac{1}{2 \ell(n-1)}ΔGRH=12−12â„“(n−1) once n,ℓn,â„“n,â„“n, \elln,â„“ have been fixed. Thus if GRH is true, for each n,ℓn,â„“n,â„“n, \elln,â„“, property Cn,ℓ(ΔGRH )Cn,ℓΔGRH C_(n,â„“)(Delta_("GRH "))\mathbf{C}_{n, \ell}\left(\Delta_{\text {GRH }}\right)Cn,â„“(ΔGRH ) holds. There has been intense interest in proving this without assuming GRH, and this will be our next topic.
3. FAMILIES OF FIELDS
So far we have considered, for each degree nnnnn, the "family" of number fields K/QK/QK//QK / \mathbb{Q}K/Q of degree nnnnn. Let us formalize this, letting Fn(X)Fn(X)F_(n)(X)\mathscr{F}_{n}(X)Fn(X) be the set of all degree nnnnn extensions KKKKK of QQQ\mathbb{Q}Q, with DK=|Disc(K/Q)|≤XDK=|Discâ¡(K/Q)|≤XD_(K)=|Disc(K//Q)| <= XD_{K}=|\operatorname{Disc}(K / \mathbb{Q})| \leq XDK=|Discâ¡(K/Q)|≤X; let Fn=Fn(∞)Fn=Fn(∞)F_(n)=F_(n)(oo)\mathscr{F}_{n}=\mathscr{F}_{n}(\infty)Fn=Fn(∞). It is helpful at this point to consider more specific families of fields of a fixed degree. For example, we could define F2−(X)F2−(X)F_(2)^(-)(X)\mathscr{F}_{2}^{-}(X)F2−(X) to be the set of imaginary quadratic fields KKKKK with DK≤XDK≤XD_(K) <= XD_{K} \leq XDK≤X, and similarly F2+(X)F2+(X)F_(2)^(+)(X)\mathscr{F}_{2}^{+}(X)F2+(X) for real quadratic fields. In general, given a transitive subgroup G⊂SnG⊂SnG subS_(n)G \subset S_{n}G⊂Sn, define the family
where all KKKKK are in a fixed algebraic closure Q¯,K~Q¯,K~bar(Q), tilde(K)\overline{\mathbb{Q}}, \tilde{K}Q¯,K~ is the Galois closure of K/QK/QK//QK / \mathbb{Q}K/Q, the Galois group is considered as a permutation group on the nnnnn embeddings of KKKKK in Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯, and the isomorphism with GGGGG is one of permutation groups. When FFF\mathscr{F}F is such a family, we define:
Property CF,ℓ(Δ)CF,â„“(Δ)C_(F,â„“)(Delta)\mathbf{C}_{\mathscr{F}, \ell}(\Delta)CF,â„“(Δ) holds if for all fields K∈F,|ClK[ℓ]|≪n,ℓ,Δ,εDKΔ+εK∈F,ClK[â„“]≪n,â„“,Δ,εDKΔ+εK inF,|Cl_(K)[â„“]|≪_(n,â„“,Delta,epsi)D_(K)^(Delta+epsi)K \in \mathscr{F},\left|\mathrm{Cl}_{K}[\ell]\right| \ll_{n, \ell, \Delta, \varepsilon} D_{K}^{\Delta+\varepsilon}K∈F,|ClK[â„“]|≪n,â„“,Δ,εDKΔ+ε for all ε>0ε>0epsi > 0\varepsilon>0ε>0.
Since Property CF,ℓ(Δ)CF,â„“(Δ)C_(F,â„“)(Delta)\mathbf{C}_{\mathscr{F}, \ell}(\Delta)CF,â„“(Δ) remains out of reach for almost all families, we also consider:
Property CF,ℓ∗(Δ)CF,ℓ∗(Δ)C_(F,â„“)^(**)(Delta)\mathbf{C}_{\mathscr{F}, \ell}^{*}(\Delta)CF,ℓ∗(Δ) holds if for almost all fields K∈F,|ClK[ℓ]|≪n,ℓ,Δ,εDKΔ+εK∈F,ClK[â„“]≪n,â„“,Δ,εDKΔ+εK inF,|Cl_(K)[â„“]|≪_(n,â„“,Delta,epsi)D_(K)^(Delta+epsi)K \in \mathscr{F},\left|\mathrm{Cl}_{K}[\ell]\right| \ll_{n, \ell, \Delta, \varepsilon} D_{K}^{\Delta+\varepsilon}K∈F,|ClK[â„“]|≪n,â„“,Δ,εDKΔ+ε for all ε>0ε>0epsi > 0\varepsilon>0ε>0. We say that a result holds for "almost all" fields in a family FFF\mathscr{F}F if the subset E(X)E(X)E(X)E(X)E(X) of possible exceptions is density zero in F(X)F(X)F(X)\mathscr{F}(X)F(X), in the sense that
|E(X)||F(X)|→0 as X→∞|E(X)||F(X)|→0 as X→∞(|E(X)|)/(|F(X)|)rarr0quad" as "X rarr oo\frac{|E(X)|}{|\mathscr{F}(X)|} \rightarrow 0 \quad \text { as } X \rightarrow \infty|E(X)||F(X)|→0 as X→∞
Here too, the first progress came for imaginary quadratic fields. Soundararajan observed that among imaginary quadratic fields with discriminant in a dyadic range [−X,−2X][−X,−2X][-X,-2X][-X,-2 X][−X,−2X], at most one can fail to satisfy |ClK[ℓ]|≪DK1/2−1/2ℓ+εClK[â„“]≪DK1/2−1/2â„“+ε|Cl_(K)[â„“]|≪D_(K)^(1//2-1//2â„“+epsi)\left|\mathrm{Cl}_{K}[\ell]\right| \ll D_{K}^{1 / 2-1 / 2 \ell+\varepsilon}|ClK[â„“]|≪DK1/2−1/2â„“+ε [82]. This verified CF2−,ℓ∗(ΔGRH)CF2−,ℓ∗ΔGRHC_(F_(2)^(-),â„“)^(**)(Delta_(GRH))\mathbf{C}_{\mathscr{F}_{2}^{-}, \ell}^{*}\left(\Delta_{\mathrm{GRH}}\right)CF2−,ℓ∗(ΔGRH) for all primes ℓâ„“â„“\ellâ„“. For ℓ=3â„“=3â„“=3\ell=3â„“=3 and quadratic fields, Wong observed that
CF2±,3∗(1/4)CF2±,3∗(1/4)C_(F_(2)^(+-),3)^(**)(1//4)\mathbf{C}_{\mathscr{F}_{2}^{ \pm}, 3}^{*}(1 / 4)CF2±,3∗(1/4) holds [96]. For any odd prime ℓâ„“â„“\ellâ„“, Heath-Brown and Pierce went below the GRH-bound, proving CF2−,ℓ∗(1/2−3/(2ℓ+2))CF2−,ℓ∗(1/2−3/(2â„“+2))C_(F_(2)^(-),â„“)^(**)(1//2-3//(2â„“+2))\mathbf{C}_{\mathscr{F}_{2}^{-}, \ell}^{*}(1 / 2-3 /(2 \ell+2))CF2−,ℓ∗(1/2−3/(2â„“+2)) [46]. They used the large sieve to show that aside from at most O(Xε)OXεO(X^(epsi))O\left(X^{\varepsilon}\right)O(Xε) exceptions, all discriminants −d∈[−X,−2X]−d∈[−X,−2X]-d in[-X,-2X]-d \in[-X,-2 X]−d∈[−X,−2X] have |ClQ(−d)[ℓ]|ClQ(−d)[â„“]|Cl_(Q(sqrt(-d)))[â„“]|\left|\mathrm{Cl}_{\mathbb{Q}(\sqrt{-d})}[\ell]\right||ClQ(−d)[â„“]|
controlled by counting the number of distinct primes p1,p2p1,p2p_(1),p_(2)p_{1}, p_{2}p1,p2 of a certain size such that (2.3) has a nontrivial integral solution (y,z)(y,z)(y,z)(y, z)(y,z). Then they showed there can be few such solutions, while averaging nontrivially over ddddd. These methods relied heavily on the explicit nature of methods for imaginary quadratic fields. Fields of higher degree need a different approach.
To apply the Ellenberg-Venkatesh criterion, we face a question such as: "Given a field, how many small primes split completely in it?" This question is very difficult in general (and is related to the Generalized Riemann Hypothesis). There is a dual question: "Given a prime, in how many fields does it split completely?" Ellenberg, Pierce, and Wood devised a method to apply the Ellenberg-Venkatesh criterion by tackling the dual question instead [33]. The idea goes like this: suppose that each prime splits completely in a positive proportion of fields in a family FFF\mathscr{F}F. Then the mean number of primes p≤xp≤xp <= xp \leq xp≤x that split completely in each field should be comparable to π(x)Ï€(x)pi(x)\pi(x)Ï€(x), and unless the primes conspire, almost all fields in FFF\mathscr{F}F should have close to the mean number of primes split completely in them. To prove that the primes cannot conspire, Ellenberg, Pierce, and Wood developed a sieve method, modeled on the Chebyshev inequality from probability.
As input the sieve requires precise counts for the cardinality
NF(X;p)=∣{K∈F(X):p splits completely in K}∣NF(X;p)=∣{K∈F(X):p splits completely in K}∣N_(F)(X;p)=∣{K inF(X):p" splits completely in "K}∣N_{\mathscr{F}}(X ; p)=\mid\{K \in \mathscr{F}(X): p \text { splits completely in } K\} \midNF(X;p)=∣{K∈F(X):p splits completely in K}∣
It also requires analogous counts NF(X;p,q)NF(X;p,q)N_(F)(X;p,q)N_{\mathscr{F}}(X ; p, q)NF(X;p,q) for when two primes p≠qp≠qp!=qp \neq qp≠q split completely in KKKKK. Suppose one can prove that for some σ>0σ>0sigma > 0\sigma>0σ>0 and τ<1Ï„<1tau < 1\tau<1Ï„<1, for all distinct primes p,qp,qp,qp, qp,q,
for a multiplicative density function δ(pq)δ(pq)delta(pq)\delta(p q)δ(pq) taking values in (0,1)(0,1)(0,1)(0,1)(0,1). Then Ellenberg, Pierce, and Wood prove that there exists Δ0>0Δ0>0Delta_(0) > 0\Delta_{0}>0Δ0>0 (depending on τ,σÏ„,σtau,sigma\tau, \sigmaÏ„,σ ) such that the mean number of primes p≤XΔ0p≤XΔ0p <= X^(Delta_(0))p \leq X^{\Delta_{0}}p≤XΔ0 that split completely in fields in F(X)F(X)F(X)\mathscr{F}(X)F(X) is comparable to π(XΔ0)Ï€XΔ0pi(X^(Delta_(0)))\pi\left(X^{\Delta_{0}}\right)Ï€(XΔ0). Moreover, there can be at most O(|F(X)|1−Δ0)O|F(X)|1−Δ0O(|F(X)|^(1-Delta_(0)))O\left(|\mathscr{F}(X)|^{1-\Delta_{0}}\right)O(|F(X)|1−Δ0) exceptional fields KKKKK in F(X)F(X)F(X)\mathscr{F}(X)F(X) such that fewer than half the mean number of primes split completely in KKKKK. Consequently, for any family FFF\mathscr{F}F for which the crucial count (3.2) can be proved, combining this sieve with the Ellenberg-Venkatesh criterion proves that CF,ℓ∗(Δ)CF,ℓ∗(Δ)C_(F,â„“)^(**)(Delta)\mathbf{C}_{\mathscr{F}, \ell}^{*}(\Delta)CF,ℓ∗(Δ) holds for every integer ℓ≥2ℓ≥2â„“ >= 2\ell \geq 2ℓ≥2, where Δ=max{12−Δ0,ΔGRH}Δ=max12−Δ0,ΔGRHDelta=max{(1)/(2)-Delta_(0),Delta_(GRH)}\Delta=\max \left\{\frac{1}{2}-\Delta_{0}, \Delta_{\mathrm{GRH}}\right\}Δ=max{12−Δ0,ΔGRH}.
For which families of fields can (3.2) be proved? Counting number fields is itself a difficult question. For each integer D≥1D≥1D >= 1D \geq 1D≥1, there are a finite number of extensions K/QK/QK//QK / \mathbb{Q}K/Q of degree nnnnn and discriminant exactly DDDDD, by Hermite's finiteness theorem [78, $4.1]. Let Nn(X)Nn(X)N_(n)(X)N_{n}(X)Nn(X) denote the number of degree nnnnn extensions K/QK/QK//QK / \mathbb{Q}K/Q with DK≤XDK≤XD_(K) <= XD_{K} \leq XDK≤X (counted up to isomorphism). A folk conjecture, sometimes associated to Linnik, states that
Nn(X)∼cnX as X→∞Nn(X)∼cnX as X→∞N_(n)(X)∼c_(n)X quad" as "X rarr ooN_{n}(X) \sim c_{n} X \quad \text { as } X \rightarrow \inftyNn(X)∼cnX as X→∞
When n=2n=2n=2n=2n=2, this is essentially equivalent to counting square-free integers (see [33, APPENDIX]). For degree n=3n=3n=3n=3n=3, this is a deep result of Davenport and Heilbronn [28]. For degree n=4n=4n=4n=4n=4, it is known by celebrated results of Cohen, Diaz y Diaz, and Olivier (counting quartic fields KKKKK with Gal(K~/Q)≃D4Galâ¡(K~/Q)≃D4Gal( tilde(K)//Q)≃D_(4)\operatorname{Gal}(\tilde{K} / \mathbb{Q}) \simeq D_{4}Galâ¡(K~/Q)≃D4 ), and Bhargava (counting non- D4D4D_(4)D_{4}D4 quartic fields) [7,20]. For degree n=5n=5n=5n=5n=5, it is known by landmark work of Bhargava [9].
The sieve method of Ellenberg, Pierce, and Wood requires an even more refined count (3.2), with prescribed local conditions and a power-saving error term with explicit dependence on p,qp,qp,qp, qp,q. Power saving error terms for Nn(X)Nn(X)N_(n)(X)N_{n}(X)Nn(X) were found for n=3n=3n=3n=3n=3 by Belabas, Bhargava, and Pomerance [5], Bhargava, Shankar, and Tsimerman [11], Taniguchi and Thorne [85]; for n=4n=4n=4n=4n=4 (non- D4D4D_(4)D_{4}D4 ) by Belabas, Bhargava, and Pomerance [5]; and for n=5n=5n=5n=5n=5 by Shankar and Tsimerman [79]. These results can be refined to prove (3.2). Ellenberg, Pierce, and Wood used this strategy to prove that when FFF\mathscr{F}F is the family of fields of degree n=2,3,4n=2,3,4n=2,3,4n=2,3,4n=2,3,4 (non- D4D4D_(4)D_{4}D4 ), or 5,CF,ℓ∗(ΔGRH)5,CF,ℓ∗ΔGRH5,C_(F,â„“)^(**)(Delta_(GRH))5, \mathbf{C}_{\mathscr{F}, \ell}^{*}\left(\Delta_{\mathrm{GRH}}\right)5,CF,ℓ∗(ΔGRH) holds for all sufficiently large primes ℓâ„“â„“\ellâ„“. (For the few remaining small ℓâ„“â„“\ellâ„“, CF,ℓ∗(Δ)CF,ℓ∗(Δ)C_(F,â„“)^(**)(Delta)\mathbf{C}_{\mathscr{F}, \ell}^{*}(\Delta)CF,ℓ∗(Δ) holds with a slightly larger Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2.) Counting quartic D4D4D_(4)D_{4}D4-fields with local conditions, ordered by discriminant, remains an interesting open problem.
The probabilistic method of Ellenberg-Pierce-Wood uses the property that the density function δ(pq)δ(pq)delta(pq)\delta(p q)δ(pq) in (3.2) is multiplicative (i.e., local conditions at ppppp and qqqqq are asymptotically independent). Frei and Widmer have adapted this approach to prove CF,ℓ∗(ΔGRH)CF,ℓ∗ΔGRHC_(F,â„“)^(**)(Delta_(GRH))\mathbf{C}_{\mathscr{F}, \ell}^{*}\left(\Delta_{\mathrm{GRH}}\right)CF,ℓ∗(ΔGRH) for all sufficiently large ℓâ„“â„“\ellâ„“, for FFF\mathscr{F}F a family of totally ramified cyclic extensions of kkkkk [40]. (That is, FFF\mathscr{F}F comprises cyclic extensions K/kK/kK//kK / kK/k of degree nnnnn in which every prime ideal of OkOkO_(k)\mathcal{O}_{k}Ok not dividing nnnnn is either unramified or totally ramified in KKKKK ). This family is chosen since the density function δ(pq)δ(pq)delta(pq)\delta(p q)δ(pq) is multiplicative. It would be interesting to investigate whether a probabilistic method can rely less strictly upon multiplicativity of the density function.
There is a great obstacle to expanding the above approach to the family of all fields of degree nnnnn when n≥6n≥6n >= 6n \geq 6n≥6. Then, even the asymptotic (3.3) is not known. For each n≥6n≥6n >= 6n \geq 6n≥6,
is the best-known bound, with c0=1.564c0=1.564c_(0)=1.564c_{0}=1.564c0=1.564, by Lemke Oliver and Thorne [61]; this improves on Couveignes [25], Ellenberg and Venkatesh [36], and Schmidt [75]. For lower bounds, in general the record is Nn(X)≫X1/2+1/nNn(X)≫X1/2+1/nN_(n)(X)≫X^(1//2+1//n)N_{n}(X) \gg X^{1 / 2+1 / n}Nn(X)≫X1/2+1/n, for all n≥7n≥7n >= 7n \geq 7n≥7 [12]. For any nnnnn divisible by p=2,3p=2,3p=2,3p=2,3p=2,3 or 5, Klüners (personal communication) has observed that Nn(X)≫XNn(X)≫XN_(n)(X)≫XN_{n}(X) \gg XNn(X)≫X, since there exists a field F/QF/QF//QF / \mathbb{Q}F/Q of degree n/pn/pn//pn / pn/p such that degree pSppSppS_(p)p S_{p}pSp-extensions of FFFFF exhibit linear asymptotics
Tackling the problem of counting primes with certain splitting conditions in a specific field via the dual problem of counting fields with certain local conditions at specific primes seems out of reach for higher degree fields. How about tackling the problem of counting primes directly?
4. COUNTING PRIMES WITH L-FUNCTIONS
The prime number theorem states that the number π(x)Ï€(x)pi(x)\pi(x)Ï€(x) of primes p≤xp≤xp <= xp \leq xp≤x satisfies π(x)∼Li(x)Ï€(x)∼Liâ¡(x)pi(x)∼Li(x)\pi(x) \sim \operatorname{Li}(x)Ï€(x)∼Liâ¡(x) as x→∞x→∞x rarr oox \rightarrow \inftyx→∞. To count small primes, or primes in short intervals, requires understanding the error term, as well as the main term. For each 1/2≤Δ<11/2≤Δ<11//2 <= Delta < 11 / 2 \leq \Delta<11/2≤Δ<1, the statement
(1.1)π(x)=Li(x)+O(xΔ+ε) for all ε>0(1.1)Ï€(x)=Liâ¡(x)+OxΔ+ε for all ε>0{:(1.1)pi(x)=Li(x)+O(x^(Delta+epsi))quad" for all "epsi > 0:}\begin{equation*}
\pi(x)=\operatorname{Li}(x)+O\left(x^{\Delta+\varepsilon}\right) \quad \text { for all } \varepsilon>0 \tag{1.1}
\end{equation*}(1.1)Ï€(x)=Liâ¡(x)+O(xΔ+ε) for all ε>0
is essentially equivalent to the statement that the Riemann zeta function ζ(s)ζ(s)zeta(s)\zeta(s)ζ(s) is zero-free for ℜ(s)>Δℜ(s)>Δℜ(s) > Delta\Re(s)>\Deltaℜ(s)>Δ [27, cн. 18]. The Riemann Hypothesis conjectures this is true for Δ=1/2Δ=1/2Delta=1//2\Delta=1 / 2Δ=1/2; it is
not known for any Δ<1Δ<1Delta < 1\Delta<1Δ<1. The best known Vinogradov-Korobov zero-free region is:
with an absolute constant C>0C>0C > 0C>0C>0 computed by Ford [37].
To count primes with a specified splitting type in a Galois extension L/QL/QL//QL / \mathbb{Q}L/Q of degree nL≥2nL≥2n_(L) >= 2n_{L} \geq 2nL≥2, consider the counting function
(4.3)πC(x,L/Q)=|{p≤x:p unramified in L,[L/Qp]=C}|(4.3)Ï€C(x,L/Q)=p≤x:p unramified in L,L/Qp=C{:(4.3){:pi_(C)(x,L//Q)=|{p <= x:p" unramified in "L,[(L//Q)/(p)]=C}|:}\begin{equation*}
\left.\pi_{\mathscr{C}}(x, L / \mathbb{Q})=\left\lvert\,\left\{p \leq x: p \text { unramified in } L,\left[\frac{L / \mathbb{Q}}{p}\right]=\mathscr{C}\right\}\right. \right\rvert\, \tag{4.3}
\end{equation*}(4.3)πC(x,L/Q)=|{p≤x:p unramified in L,[L/Qp]=C}|
in which [L/Qp]L/Qp[(L//Q)/(p)]\left[\frac{L / \mathbb{Q}}{p}\right][L/Qp] is the Artin symbol and CCC\mathscr{C}C is any fixed conjugacy class in G=Gal(L/Q)G=Galâ¡(L/Q)G=Gal(L//Q)G=\operatorname{Gal}(L / \mathbb{Q})G=Galâ¡(L/Q). For example, when L=Q(e2πi/q)L=Qe2Ï€i/qL=Q(e^(2pi i//q))L=\mathbb{Q}\left(e^{2 \pi i / q}\right)L=Q(e2Ï€i/q), this can be used to count primes in a fixed residue class modulo qqqqq. Or, for example, for any Galois extension L/QL/QL//QL / \mathbb{Q}L/Q, when C={Id}C={Id}C={Id}\mathscr{C}=\{\mathrm{Id}\}C={Id}, this counts primes that split completely in LLLLL. By the celebrated Chebotarev density theorem [88],
(4.4)πC(x,L/Q)∼|C||G|Li(x), as x→∞(4.4)Ï€C(x,L/Q)∼|C||G|Liâ¡(x), as x→∞{:(4.4)pi_(C)(x","L//Q)∼(|C|)/(|G|)Li(x)","quad" as "x rarr oo:}\begin{equation*}
\pi_{\mathscr{C}}(x, L / \mathbb{Q}) \sim \frac{|\mathscr{C}|}{|G|} \operatorname{Li}(x), \quad \text { as } x \rightarrow \infty \tag{4.4}
\end{equation*}(4.4)Ï€C(x,L/Q)∼|C||G|Liâ¡(x), as x→∞
But just as for π(x)Ï€(x)pi(x)\pi(x)Ï€(x), to count small primes accurately requires more quantitative information. A central goal is to prove an asymptotic for πC(x,L/Q)Ï€C(x,L/Q)pi_(C)(x,L//Q)\pi_{\mathscr{C}}(x, L / \mathbb{Q})Ï€C(x,L/Q) that is valid for xxxxx very small relative to DL=|DiscL/Q|DL=|Discâ¡L/Q|D_(L)=|Disc L//Q|D_{L}=|\operatorname{Disc} L / \mathbb{Q}|DL=|Discâ¡L/Q|, and with an effective error term. This requires exhibiting a zero-free region for the Dedekind zeta function ζL(s)ζL(s)zeta_(L)(s)\zeta_{L}(s)ζL(s). This is more complicated than (4.2), due to the possibility of an exceptional Landau-Siegel zero: within the region
ζL(σ+it)ζL(σ+it)zeta_(L)(sigma+it)\zeta_{L}(\sigma+i t)ζL(σ+it) can contain at most one (real, simple) zero, denoted β0β0beta_(0)\beta_{0}β0 if it exists. (As observed by Heilbronn and generalized by Stark, if β0β0beta_(0)\beta_{0}β0 exists then it must "come from" a quadratic field, in the sense that LLLLL contains a quadratic subfield FFFFF with ζF(β0)=0[47,83]ζFβ0=0[47,83]zeta_(F)(beta_(0))=0[47,83]\zeta_{F}\left(\beta_{0}\right)=0[47,83]ζF(β0)=0[47,83].)
Lagarias and Odlyzko used the zero-free region (4.5) to prove there exist absolute, computable constants C1,C2C1,C2C_(1),C_(2)C_{1}, C_{2}C1,C2 such that for all x≥exp(10nL(logDL)2)x≥expâ¡10nLlogâ¡DL2x >= exp(10n_(L)(log D_(L))^(2))x \geq \exp \left(10 n_{L}\left(\log D_{L}\right)^{2}\right)x≥expâ¡(10nL(logâ¡DL)2),
in which the β0β0beta_(0)\beta_{0}β0 term is present only if β0β0beta_(0)\beta_{0}β0 exists (see [60], and Serre [77]). This was the first effective Chebotarev density theorem. It can be difficult to apply to questions of interest because of the mysterious β0β0beta_(0)\beta_{0}β0 term, and because xxxxx must be a large power of DLDLD_(L)D_{L}DL (certainly at least x≥DL10nLx≥DL10nLx >= D_(L)^(10n_(L))x \geq D_{L}^{10 n_{L}}x≥DL10nL ). In contrast, to apply the Ellenberg-Venkatesh criterion to a field KKKKK of degree nnnnn, we aim to exhibit primes p<DKηp<DKηp < D_(K)^(eta)p<D_{K}^{\eta}p<DKη that split completely in the Galois closure K~K~tilde(K)\tilde{K}K~ (and hence in KKKKK ), with η≈1/(2ℓ(n−1))→0η≈1/(2â„“(n−1))→0eta~~1//(2â„“(n-1))rarr0\eta \approx 1 /(2 \ell(n-1)) \rightarrow 0η≈1/(2â„“(n−1))→0 as n,ℓ→∞n,ℓ→∞n,â„“rarr oon, \ell \rightarrow \inftyn,ℓ→∞. (These primes are even smaller relative to DK~DK~D_( tilde(K))D_{\tilde{K}}DK~, since DK|G|/n≪GDK~≪GDK|G|/2DK|G|/n≪GDK~≪GDK|G|/2D_(K)^(|G|//n)≪_(G)D_( tilde(K))≪_(G)D_(K)^(|G|//2)D_{K}^{|G| / n} \ll_{G} D_{\tilde{K}} \ll_{G} D_{K}^{|G| / 2}DK|G|/n≪GDK~≪GDK|G|/2, where G=Gal(K~/Q)G=Galâ¡(K~/Q)G=Gal( tilde(K)//Q)G=\operatorname{Gal}(\tilde{K} / \mathbb{Q})G=Galâ¡(K~/Q) [72].)
If GRH holds for ζL(s)ζL(s)zeta_(L)(s)\zeta_{L}(s)ζL(s), then ζL(s)ζL(s)zeta_(L)(s)\zeta_{L}(s)ζL(s) is zero-free for ℜ(s)>1/2ℜ(s)>1/2ℜ(s) > 1//2\Re(s)>1 / 2ℜ(s)>1/2, and Lagarias and Odlyzko improve (4.6) in three ways: (i) it is valid for x≥2x≥2x >= 2x \geq 2x≥2; (ii) the β0β0beta_(0)\beta_{0}β0 term is not present; (iii) the remaining error term is O(x1/2log(DLxnL))Ox1/2logâ¡DLxnLO(x^(1//2)log(D_(L)x^(n_(L))))O\left(x^{1 / 2} \log \left(D_{L} x^{n_{L}}\right)\right)O(x1/2logâ¡(DLxnL)). Properties (i) and (ii) show that for every η>0η>0eta > 0\eta>0η>0, for every degree nnnnn extension K/QK/QK//QK / \mathbb{Q}K/Q with DKDKD_(K)D_{K}DK sufficiently large, at least ≫π(DKη)≫πDKη≫pi(D_(K)^(eta))\gg \pi\left(D_{K}^{\eta}\right)≫π(DKη) primes p≤DKηp≤DKηp <= D_(K)^(eta)p \leq D_{K}^{\eta}p≤DKη split completely in the Galois closure K~K~tilde(K)\tilde{K}K~ (and hence in KKKKK ). When input into
the Ellenberg-Venkatesh criterion, this is the source of the GRH-bound (2.5) for all integers ℓ≥2ℓ≥2â„“ >= 2\ell \geq 2ℓ≥2.
Here is a central goal: improve the Chebotarev density theorem (4.6) without assuming GRH, so that (i') for any η>0η>0eta > 0\eta>0η>0 it is valid for xxxxx as small as x≥DLηx≥DLηx >= D_(L)^(eta)x \geq D_{L}^{\eta}x≥DLη (for all DLDLD_(L)D_{L}DL sufficiently large) and (ii) the β0β0beta_(0)\beta_{0}β0 term is not present. (For many applications, the final error term in (4.6) suffices as is.) If this held for L=K~L=K~L= tilde(K)L=\tilde{K}L=K~ the Galois closure of a field KKKKK, the Ellenberg-Venkatesh criterion would imply the GRH-bound (2.5) for ℓâ„“â„“\ellâ„“-torsion in ClKClKCl_(K)\mathrm{Cl}_{K}ClK for all integers ℓ≥2ℓ≥2â„“ >= 2\ell \geq 2ℓ≥2, without assuming GRH. Recently, Pierce, Turnage-Butterbaugh, and Wood showed that the key improvements (i') and (ii) hold if for some 0<δ≤1/4,ζL(s)/ζ(s)0<δ≤1/4,ζL(s)/ζ(s)0 < delta <= 1//4,zeta_(L)(s)//zeta(s)0<\delta \leq 1 / 4, \zeta_{L}(s) / \zeta(s)0<δ≤1/4,ζL(s)/ζ(s) is zero-free for s=σ+its=σ+its=sigma+its=\sigma+i ts=σ+it in the box
Proving this for any particular LLLLL-function ζL(s)/ζ(s)ζL(s)/ζ(s)zeta_(L)(s)//zeta(s)\zeta_{L}(s) / \zeta(s)ζL(s)/ζ(s) of interest is out of reach. Instead, it can be productive to study a family of LLLLL-functions. In particular, if F=Fn(G;X)F=Fn(G;X)F=F_(n)(G;X)\mathscr{F}=\mathscr{F}_{n}(G ; X)F=Fn(G;X) is a family of degree nnnnn fields with fixed Galois group of the Galois closure, property CF,ℓ∗(ΔGRH)CF,ℓ∗ΔGRHC_(F,â„“)^(**)(Delta_(GRH))\mathbf{C}_{\mathscr{F}, \ell}^{*}\left(\Delta_{\mathrm{GRH}}\right)CF,ℓ∗(ΔGRH) will follow (for all integers ℓ≥2ℓ≥2â„“ >= 2\ell \geq 2ℓ≥2 ) if it is true for almost all fields K∈Fn(G;X)K∈Fn(G;X)K inF_(n)(G;X)K \in \mathscr{F}_{n}(G ; X)K∈Fn(G;X), that ζK~(s)/ζ(s)ζK~(s)/ζ(s)zeta_( tilde(K))(s)//zeta(s)\zeta_{\tilde{K}}(s) / \zeta(s)ζK~(s)/ζ(s) is zero-free in the box (4.7). This was the strategy Pierce, Turnage-Butterbaugh, and Wood developed in [72], which we will now briefly sketch.
4.1. Families of LLLLL-functions
There is a long history of estimating the density of zeroes within a certain region, for a family of LLLLL-functions. If we can show there are fewer possible zeroes in the region than there are LLLLL-functions in the family, then some of the LLLLL-functions must be zero-free in that region. We single out a result of Kowalski and Michel, who used the large sieve to prove a zero density result for families of cuspidal automorphic LLLLL-functions [56]. In particular, for suitable families, their result implies that almost all LLLLL-functions in the family must be zero-free in a box analogous to (4.7).
There are two fundamental barriers to applying this to our problem of interest: the representation underlying ζK~(s)/ζ(s)ζK~(s)/ζ(s)zeta_( tilde(K))(s)//zeta(s)\zeta_{\tilde{K}}(s) / \zeta(s)ζK~(s)/ζ(s) is not always cuspidal, and it is not always known to be automorphic. Suppose GGGGG has irreducible complex representations ρ0,ρ1,…,ρrÏ0,Ï1,…,Ïrrho_(0),rho_(1),dots,rho_(r)\rho_{0}, \rho_{1}, \ldots, \rho_{r}Ï0,Ï1,…,Ïr, with ρ0Ï0rho_(0)\rho_{0}Ï0 the trivial representation. Then for K∈Fn(G;X),ζK~K∈Fn(G;X),ζK~K inF_(n)(G;X),zeta_( tilde(K))K \in \mathscr{F}_{n}(G ; X), \zeta_{\tilde{K}}K∈Fn(G;X),ζK~ is a product of Artin LLLLL-functions,
The Artin (holomorphy) conjecture posits that for each nontrivial irreducible representation ρj,L(s,ρj,K~/Q)Ïj,Ls,Ïj,K~/Qrho_(j),L(s,rho_(j),( tilde(K))//Q)\rho_{j}, L\left(s, \rho_{j}, \tilde{K} / \mathbb{Q}\right)Ïj,L(s,Ïj,K~/Q) is entire. The (strong) Artin conjecture posits that for each nontrivial irreducible representation ρjÏjrho_(j)\rho_{j}Ïj, there is an associated cuspidal automorphic representation πK~,jÏ€K~,jpi_( tilde(K),j)\pi_{\tilde{K}, j}Ï€K~,j of GL(mj)/QGLmj/QGL(m_(j))//Q\mathrm{GL}\left(m_{j}\right) / \mathbb{Q}GL(mj)/Q, and L(s,πK~,j)=L(s,ρj,K~/Q)Ls,Ï€K~,j=Ls,Ïj,K~/QL(s,pi_( tilde(K),j))=L(s,rho_(j),( tilde(K))//Q)L\left(s, \pi_{\tilde{K}, j}\right)=L\left(s, \rho_{j}, \tilde{K} / \mathbb{Q}\right)L(s,Ï€K~,j)=L(s,Ïj,K~/Q). This is known for certain types of representations of certain groups, but otherwise is a deep open problem (see recent work in [19]). For the moment, we will proceed by assuming the strong conjecture. Then the factorization (4.8) naturally slices the family ζK~(s)/ζ(s)ζK~(s)/ζ(s)zeta_( tilde(K))(s)//zeta(s)\zeta_{\tilde{K}}(s) / \zeta(s)ζK~(s)/ζ(s), as KKKKK varies over Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X), into rrrrr families L1(X),L2(X),…,Lr(X)L1(X),L2(X),…,Lr(X)L_(1)(X),L_(2)(X),dots,L_(r)(X)\mathscr{L}_{1}(X), \mathscr{L}_{2}(X), \ldots, \mathscr{L}_{r}(X)L1(X),L2(X),…,Lr(X), where each Lj(X)Lj(X)L_(j)(X)\mathscr{L}_{j}(X)Lj(X) is the set of cuspidal automorphic representations πK~,jÏ€K~,jpi_( tilde(K),j)\pi_{\tilde{K}, j}Ï€K~,j associated to the representation ρjÏjrho_(j)\rho_{j}Ïj. Kowalski and Michel's result applies
to each family Lj(X)Lj(X)L_(j)(X)\mathscr{L}_{j}(X)Lj(X) individually. This proves that every representation π∈Lj(X)π∈Lj(X)pi inL_(j)(X)\pi \in \mathscr{L}_{j}(X)π∈Lj(X) has associated LLLLL-function L(s,π)L(s,Ï€)L(s,pi)L(s, \pi)L(s,Ï€) being zero-free in the box (4.7)—except for a possible subset of "bad" representations πÏ€pi\piÏ€, of density zero in Lj(X)Lj(X)L_(j)(X)\mathscr{L}_{j}(X)Lj(X), for which L(s,π)L(s,Ï€)L(s,pi)L(s, \pi)L(s,Ï€) could have a zero in the box. (Of course, no such zero exists if GRH is true, but we are not assuming GRH.)
Now a crucial difficulty arises: if there were a "bad" representation π∈Lj(X)π∈Lj(X)pi inL_(j)(X)\pi \in \mathscr{L}_{j}(X)π∈Lj(X), in how many products (4.8) could it appear, as KKKKK varies over Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) ? Each field KKKKK for which the "bad" factor L(s,π)L(s,Ï€)L(s,pi)L(s, \pi)L(s,Ï€) appears could have a zero of ζK~(s)/ζ(s)ζK~(s)/ζ(s)zeta_( tilde(K))(s)//zeta(s)\zeta_{\tilde{K}}(s) / \zeta(s)ζK~(s)/ζ(s) in (4.7). Thus the crucial question is: for a fixed nontrivial irreducible representation ρÏrho\rhoÏ of GGGGG, how many fields K1,K2∈Fn(G;X)K1,K2∈Fn(G;X)K_(1),K_(2)inF_(n)(G;X)K_{1}, K_{2} \in \mathscr{F}_{n}(G ; X)K1,K2∈Fn(G;X) have L(s,ρ,K~1/Q)=L(s,ρ,K~2/Q)Ls,Ï,K~1/Q=Ls,Ï,K~2/QL(s,rho, tilde(K)_(1)//Q)=L(s,rho, tilde(K)_(2)//Q)L\left(s, \rho, \tilde{K}_{1} / \mathbb{Q}\right)=L\left(s, \rho, \tilde{K}_{2} / \mathbb{Q}\right)L(s,Ï,K~1/Q)=L(s,Ï,K~2/Q) ? This can be stated a different way. Given a subgroup HHHHH of GGGGG, let K~HK~Htilde(K)^(H)\tilde{K}^{H}K~H denote the subfield of K~K~tilde(K)\tilde{K}K~ fixed by HHHHH. It turns out that the question can be transformed into: how many fields K1,K2∈Fn(G;X)K1,K2∈Fn(G;X)K_(1),K_(2)inF_(n)(G;X)K_{1}, K_{2} \in \mathscr{F}_{n}(G ; X)K1,K2∈Fn(G;X) have K~1Ker(ρ)=K~2Ker(ρ)K~1Kerâ¡(Ï)=K~2Kerâ¡(Ï)tilde(K)_(1)^(Ker(rho))= tilde(K)_(2)^(Ker(rho))\tilde{K}_{1}^{\operatorname{Ker}(\rho)}=\tilde{K}_{2}^{\operatorname{Ker}(\rho)}K~1Kerâ¡(Ï)=K~2Kerâ¡(Ï) ? Let us call this a collision. If a positive proportion of fields in Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) can collide for ρjÏjrho_(j)\rho_{j}Ïj, then via the factorization (4.8), the possible existence of even one "bad" element in Lj(X)Lj(X)L_(j)(X)\mathscr{L}_{j}(X)Lj(X) could allow a positive proportion of the functions ζK~(s)/ζ(s)ζK~(s)/ζ(s)zeta_( tilde(K))(s)//zeta(s)\zeta_{\tilde{K}}(s) / \zeta(s)ζK~(s)/ζ(s) to have a zero in (4.7). In particular, then this approach would fail to prove CF,ℓ∗(ΔGRH)CF,ℓ∗ΔGRHC_(F,â„“)^(**)(Delta_(GRH))\mathbf{C}_{\mathscr{F}, \ell}^{*}\left(\Delta_{\mathrm{GRH}}\right)CF,ℓ∗(ΔGRH) for the family F=Fn(G;X)F=Fn(G;X)F=F_(n)(G;X)\mathscr{F}=\mathscr{F}_{n}(G ; X)F=Fn(G;X). To rule this out, we aim to show that for each nontrivial irreducible representation ρjÏjrho_(j)\rho_{j}Ïj of GGGGG, collisions are rare.
We define the "collision problem" for the family Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) : how big is
Here the maximum is over the nontrivial irreducible representations ρÏrho\rhoÏ of GGGGG with Ker(ρ)Kerâ¡(Ï)Ker(rho)\operatorname{Ker}(\rho)Kerâ¡(Ï) a proper normal subgroup of GGGGG. Suppose for a particular family Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X), the collisions (4.9) number at most ≪Xα≪Xα≪X^(alpha)\ll X^{\alpha}≪Xα. Then the strategy sketched here ultimately shows that aside from at most ≪Xα+ε≪Xα+ε≪X^(alpha+epsi)\ll X^{\alpha+\varepsilon}≪Xα+ε exceptional fields (for any ε>0ε>0epsi > 0\varepsilon>0ε>0 ), every field in K∈Fn(G;X)K∈Fn(G;X)K inF_(n)(G;X)K \in \mathscr{F}_{n}(G ; X)K∈Fn(G;X) has the property that an improved Chebotarev density theorem with properties (i') and (ii) holds for its Galois closure K~K~tilde(K)\tilde{K}K~. If we can prove simultaneously that |Fn(G;X)|≫XβFn(G;X)≫Xβ|F_(n)(G;X)|≫X^(beta)\left|\mathscr{F}_{n}(G ; X)\right| \gg X^{\beta}|Fn(G;X)|≫Xβ for some β>αβ>αbeta > alpha\beta>\alphaβ>α, then the improved Chebotarev density theorem holds for almost all fields in the family. Consequently, we would obtain property CF,ℓ∗(ΔGRH)CF,ℓ∗ΔGRHC_(F,â„“)^(**)(Delta_(GRH))\mathbf{C}_{\mathscr{F}, \ell}^{*}\left(\Delta_{\mathrm{GRH}}\right)CF,ℓ∗(ΔGRH), for all integers ℓ≥2ℓ≥2â„“ >= 2\ell \geq 2ℓ≥2.
Thus the goal of bounding ℓâ„“â„“\ellâ„“-torsion in class groups of fields in the family Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) has been transformed into a question of counting how often certain fields share a subfield. For which families can the collision problem (4.9) be controlled? For some groups, the number of collisions can be ≫|Fn(G;X)|≫Fn(G;X)≫|F_(n)(G;X)|\gg\left|\mathscr{F}_{n}(G ; X)\right|≫|Fn(G;X)| (for example, G=Z/4Z)G=Z/4Z{:G=Z//4Z)\left.G=\mathbb{Z} / 4 \mathbb{Z}\right)G=Z/4Z). On the other hand, if GGGGG is a simple group, or if all nontrivial irreducible representations of GGGGG are faithful, the number of collisions is ≪1≪1≪1\ll 1≪1 (but a lower bound |Fn(G;X)|≫XβFn(G;X)≫Xβ|F_(n)(G;X)|≫X^(beta)\left|\mathscr{F}_{n}(G ; X)\right| \gg X^{\beta}|Fn(G;X)|≫Xβ for some β>0β>0beta > 0\beta>0β>0 may not be known, yet). In general, controlling the collision problem is difficult.
One idea is to restrict attention to an advantageously chosen subfamily of fields, call it Fn∗(G;X)⊂Fn(G;X)Fn∗(G;X)⊂Fn(G;X)F_(n)^(**)(G;X)subF_(n)(G;X)\mathscr{F}_{n}^{*}(G ; X) \subset \mathscr{F}_{n}(G ; X)Fn∗(G;X)⊂Fn(G;X). To bound (4.9) within a subfamily it suffices to count
Here HHHHH ranges over the proper normal subgroups of GGGGG that appear as the kernel of some nontrivial irreducible representation. For some groups GGGGG, if Fn∗(G;X)Fn∗(G;X)F_(n)^(**)(G;X)\mathscr{F}_{n}^{*}(G ; X)Fn∗(G;X) is defined appropriately, this can be further transformed into counting number fields with fixed discriminant.
Let us see how this goes in the example G=SnG=SnG=S_(n)G=S_{n}G=Sn with n=3n=3n=3n=3n=3 or n≥5n≥5n >= 5n \geq 5n≥5, so that AnAnA_(n)A_{n}An is the only nontrivial proper normal subgroup (the kernel of the sign representation). Consider the subfamily Fn∗(Sn;X)Fn∗Sn;XF_(n)^(**)(S_(n);X)\mathscr{F}_{n}^{*}\left(S_{n} ; X\right)Fn∗(Sn;X) of fields with square-free discriminant. (These are a positive proportion of all degree nSnnSnnS_(n)n S_{n}nSn-fields for n≤5n≤5n <= 5n \leq 5n≤5 and conjecturally so for n≥6n≥6n >= 6n \geq 6n≥6.) Then for H=AnH=AnH=A_(n)H=A_{n}H=An and FFFFF a fixed quadratic field, it can be shown that any field KKKKK counted in (4.10) must have the property that DK=DFDK=DFD_(K)=D_(F)D_{K}=D_{F}DK=DF (up to some easily controlled behavior of wildly ramified primes). Under this very strong identity of discriminants, (4.10) is dominated by
This strategy transforms the collision problem into counting fields of fixed discriminant.
For certain other groups GGGGG, (4.10) can also be dominated by a quantity analogous to (4.11) if the subfamily Fn∗(G;X)Fn∗(G;X)F_(n)^(**)(G;X)\mathscr{F}_{n}^{*}(G ; X)Fn∗(G;X) is defined by specifying that each prime that is tamely ramified in KKKKK has its inertia group generated by an element in a carefully chosen conjugacy class III\mathscr{I}I of GGGGG. For such a group GGGGG, the final step in this strategy for proving improved Chebotarev density theorems for almost all fields in the family Fn∗(G;X)Fn∗(G;X)F_(n)^(**)(G;X)\mathscr{F}_{n}^{*}(G ; X)Fn∗(G;X) is to bound (4.11). If |Fn∗(G;X)|≫XβFn∗(G;X)≫Xβ|F_(n)^(**)(G;X)|≫X^(beta)\left|\mathscr{F}_{n}^{*}(G ; X)\right| \gg X^{\beta}|Fn∗(G;X)|≫Xβ is known, it suffices to prove (4.11) is ≪Xα≪Xα≪X^(alpha)\ll X^{\alpha}≪Xα for some α<βα<βalpha < beta\alpha<\betaα<β. In general, counting number fields with fixed discriminant is very difficult-we will return to this problem later. But for some families Fn∗(G;X)Fn∗(G;X)F_(n)^(**)(G;X)\mathscr{F}_{n}^{*}(G ; X)Fn∗(G;X), (4.11) can be controlled sufficiently well, relative to a known lower bound for |Fn∗(G;X)|Fn∗(G;X)|F_(n)^(**)(G;X)|\left|\mathscr{F}_{n}^{*}(G ; X)\right||Fn∗(G;X)|.
This is the strategy developed by Pierce, Turnage-Butterbaugh, and Wood in [72]. The result is an improved Chebotarev density theorem, with properties (i') and (ii), that holds unconditionally for almost all fields in the following families: (a) Fp(Cp;X)FpCp;XF_(p)(C_(p);X)\mathscr{F}_{p}\left(C_{p} ; X\right)Fp(Cp;X) cyclic extensions of any prime degree; (b) Fn∗(Cn;X)Fn∗Cn;XF_(n)^(**)(C_(n);X)\mathscr{F}_{n}^{*}\left(C_{n} ; X\right)Fn∗(Cn;X) totally ramified cyclic extensions of any degree n≥2n≥2n >= 2n \geq 2n≥2; (c) Fp∗(Dp;X)Fp∗Dp;XF_(p)^(**)(D_(p);X)\mathscr{F}_{p}^{*}\left(D_{p} ; X\right)Fp∗(Dp;X) prime degree dihedral extensions, III\mathscr{I}I being the class of order 2 elements; (d) Fn∗(Sn;X)Fn∗Sn;XF_(n)^(**)(S_(n);X)\mathscr{F}_{n}^{*}\left(S_{n} ; X\right)Fn∗(Sn;X) fields of square-free discriminant, n=3n=3n=3n=3n=3, 4 ; and (e) F4∗(A4;X)F4∗A4;XF_(4)^(**)(A_(4);X)\mathscr{F}_{4}^{*}\left(A_{4} ; X\right)F4∗(A4;X), III\mathscr{I}I being either class of order 3 elements. Conditional on the strong Artin conjecture, they proved the improved Chebotarev density theorem also holds for almost all fields in the following families: (f) F5∗(S5;X)F5∗S5;XF_(5)^(**)(S_(5);X)\mathscr{F}_{5}^{*}\left(S_{5} ; X\right)F5∗(S5;X) quintic fields of square-free discriminant; and (g) Fn(An;X)FnAn;XF_(n)(A_(n);X)\mathscr{F}_{n}\left(A_{n} ; X\right)Fn(An;X), for all n≥5n≥5n >= 5n \geq 5n≥5. (There are other families, such as Fn∗(Sn;X)Fn∗Sn;XF_(n)^(**)(S_(n);X)\mathscr{F}_{n}^{*}\left(S_{n} ; X\right)Fn∗(Sn;X) for n≥6n≥6n >= 6n \geq 6n≥6, to which the strategy applies, but the current upper bound known for (4.11) is larger than the known lower bound for |Fn∗(Sn;X)|Fn∗Sn;X|F_(n)^(**)(S_(n);X)|\left|\mathscr{F}_{n}^{*}\left(S_{n} ; X\right)\right||Fn∗(Sn;X)|.) As a consequence, Pierce, Turnage-Butterbaugh, and Wood proved for each family (a)-(e) that CF,n∗(ΔGRH )CF,n∗ΔGRH C_(F,n)^(**)(Delta_("GRH "))\mathbf{C}_{\mathscr{F}, n}^{*}\left(\Delta_{\text {GRH }}\right)CF,n∗(ΔGRH ) holds unconditionally for all integers ℓ≥2ℓ≥2â„“ >= 2\ell \geq 2ℓ≥2, and it holds for each family (f)-(g) under the strong Artin conjecture. This was the first time such a result was proved for families of fields of arbitrarily large degree.
4.2. Further developments
Since the work outlined above, many interesting new developments have followed, relating to zero density results for families of LLLLL-functions, Chebotarev density theorems for families of fields, and ℓâ„“â„“\ellâ„“-torsion in class groups of fields in specific families.
First, there has been renewed interest in zero density results for families of LLLLL-functions, concerning potential zeroes in regions close to the line ℜ(s)=1ℜ(s)=1ℜ(s)=1\Re(s)=1ℜ(s)=1, and extending the perspective of Kowalski and Michel [56]; see, for example, [18,49,87].
Second, several new strategies have focused on the problem of proving effective Chebotarev density theorems for almost all fields in a family. The work in [72] raised several desiderata. Some groups GGGGG have the property that no ramification restriction exists that allows the "collision problem" in the form (4.10) to be transformed into a "discriminant multiplicity problem" in the form (4.11). For example, this occurs for any noncyclic abelian group, or D4D4D_(4)D_{4}D4. These cases remain open; instead, An recently proved a Chebotarev density theorem for almost all fields in a family of quartic D4D4D_(4)D_{4}D4-fields associated to a fixed biquadratic field [2]. Another significant desideratum was to remove the dependence on the strong Artin conjecture. Thorner and Zaman recently achieved this, by proving a zero density estimate directly for Dedekind zeta functions, without passing through the factorization (4.8) [86]. But that work is still explicitly conditional on the ability to control a collision problem similar to (4.9), for which the best known strategy is still the approach of [72].
Most recently, the collision problem has been bypassed for certain groups GGGGG by interesting new work of Lemke Oliver, Thorner, and Zaman [62]. Their key idea when studying fields in a family Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) is to prove a zero-free region not for ζK~/ζζK~/ζzeta_( tilde(K))//zeta\zeta_{\tilde{K}} / \zetaζK~/ζ but for ζK~/ζK~NζK~/ζK~Nzeta_( tilde(K))//zeta_( tilde(K)^(N))\zeta_{\tilde{K}} / \zeta_{\tilde{K}^{N}}ζK~/ζK~N where NNNNN is a nontrivial normal subgroup of GGGGG. This allows them to replace a collision problem like (4.9) by an "intersection multiplicity problem," bounding
The number of exceptional fields, for which a desired Chebotarev-type theorem cannot be verified, is then dominated by (4.12) (up to XεXεX^(epsi)X^{\varepsilon}Xε ). This is advantageous if GGGGG has a unique minimal nontrivial normal subgroup NNNNN, so that (4.12) is ≪1≪1≪1\ll 1≪1. But as a trade-off, one no longer obtains an effective Chebotarev density theorem for each conjugacy class CCC\mathscr{C}C in GGGGG.
Let πK(x)Ï€K(x)pi_(K)(x)\pi_{K}(x)Ï€K(x) count prime ideals p⊂OKp⊂OKpsubO_(K)\mathfrak{p} \subset \mathcal{O}_{K}p⊂OK with ℜK/Qp≤xℜK/Qp≤xℜ_(K//Q)p <= x\Re_{K / \mathbb{Q}} \mathfrak{p} \leq xℜK/Qp≤x. Let FFF\mathscr{F}F represent either of the two following families: degree ppppp fields K/QK/QK//QK / \mathbb{Q}K/Q for ppppp prime, or degree nSnnSnnS_(n)n S_{n}nSn-fields K/QK/QK//QK / \mathbb{Q}K/Q, for any n≥2n≥2n >= 2n \geq 2n≥2. Lemke Oliver, Thorner, and Zaman prove that except for at most ≪Xε≪Xε≪X^(epsi)\ll X^{\varepsilon}≪Xε exceptional fields, every K∈F(X)K∈F(X)K inF(X)K \in \mathscr{F}(X)K∈F(X) has |πK(x)−π(x)|≤C1xexp(−C2logx)Ï€K(x)−π(x)≤C1xexpâ¡âˆ’C2logâ¡x|pi_(K)(x)-pi(x)| <= C_(1)x exp(-C_(2)sqrt(log x))\left|\pi_{K}(x)-\pi(x)\right| \leq C_{1} x \exp \left(-C_{2} \sqrt{\log x}\right)|Ï€K(x)−π(x)|≤C1xexpâ¡(−C2logâ¡x) for every x≥(logDK)C3(n,ε)x≥logâ¡DKC3(n,ε)x >= (log D_(K))^(C_(3)(n,epsi))x \geq\left(\log D_{K}\right)^{C_{3}(n, \varepsilon)}x≥(logâ¡DK)C3(n,ε). In either family FFF\mathscr{F}F, they obtain results on ℓâ„“â„“\ellâ„“-torsion by applying the Ellenberg-Venkatesh criterion using prime ideals of degree 1 . If πK∗(x)Ï€K∗(x)pi_(K)^(**)(x)\pi_{K}^{*}(x)Ï€K∗(x) counts only prime ideals of degree 1 , then πK∗(x)=πK(x)+On(x)Ï€K∗(x)=Ï€K(x)+On(x)pi_(K)^(**)(x)=pi_(K)(x)+O_(n)(sqrtx)\pi_{K}^{*}(x)=\pi_{K}(x)+O_{n}(\sqrt{x})Ï€K∗(x)=Ï€K(x)+On(x), so the above result exhibits many small prime ideals of degree 1 . Thus for either family, CF,n∗(ΔGRH)CF,n∗ΔGRHC_(F,n)^(**)(Delta_(GRH))\mathbf{C}_{\mathscr{F}, n}^{*}\left(\Delta_{\mathrm{GRH}}\right)CF,n∗(ΔGRH) holds unconditionally for all ℓâ„“â„“\ellâ„“ (and the exceptional set is very small). (They also exhibit infinitely many degree nSnnSnnS_(n)n S_{n}nSn-fields KKKKK with ClKClKCl_(K)\mathrm{Cl}_{K}ClK as large as possible, but |ClK[ℓ]|ClK[â„“]|Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right|